8.1. (xn).

h‚“yˆ~†Œ
p‚†Žz• ~Ž†…‰›Š
mŽ†•‰Œ| ‡~† •~•†‡z• †•†™‘„‘‚•
mogpjmp
f‚–ŽŒš‰‚ Œ•Œ†~•{•Œ‘‚ ~‡ŒˆŒ’…|~ (xn ) qŒ •š‰•ŒˆŒ
P+∞
x1 + x2 + · · · + xn + · · · .
{
n=1 xn
P
ŒŠŒ‰yƒ‚‘~† •‚†Žy ‡~† „ ‘†‰{ ‘Œ’ ~Š ~’‘{ ’•yŽ”‚† ŒŠŒ‰yƒ‚‘~† y…ŽŒ†•‰~ ‘„• •‚†Žy• +∞
n=1 xn ‡~†
~’‘{ „ ‘†‰{ ‡~…ŒŽ|ƒ‚‘~† ‰‚ ‘„Š •~Ž~‡y‘– •†~•†‡~•|~
p”„‰~‘|ƒŒ’‰‚ ‘~ •†~•Œ”†‡y ~…ŽŒ|•‰~‘~ s1 = x1 s2 = x1 + x2 s3 = x1 + x2 + x3 ‡~† €‚Š†‡™‘‚Ž~
sn = x 1 + · · · + x n
€†~ ‡y…‚ n.
P
m sn ŒŠŒ‰yƒ‚‘~† n Œ•‘™ ‰‚Ž†‡™ y…ŽŒ†•‰~ ‘„• •‚†Žy• +∞
n=1 xn ‡~† „ ~‡ŒˆŒ’…|~ (sn ) ŒŠŒ‰yƒ‚‘~†
~‡ŒˆŒ’…|~ ‘–Š ‰‚Ž†‡›Š ~…ŽŒ†•‰y‘–Š ‘„• •‚†Žy•
P
_Š „ ~‡ŒˆŒ’…|~ (sn ) z”‚† ™Ž†Œ •„ˆ~•{ ~Š sn → s ∈ R ‘™‘‚ ˆz‰‚ ™‘† „ •‚†Žy +∞
n=1 xn z”‚†
y…ŽŒ†•‰~ ‡~† –• y…ŽŒ†•‰~ ‘„• •‚†Žy• …‚–ŽŒš‰‚ ‘Œ ™Ž†Œ s ‡~† €Žy“Œ’‰‚
P+∞
{
x1 + x2 + · · · + xn + · · · = s.
n=1 xn = s
c†•†‡›‘‚Ž~ ~Š s ∈ R ˆz‰‚ ™‘† „ •‚†Žy •’€‡ˆ|Š‚† •‘ŒŠ s ‡~† ~Š s = +∞ { s = −∞ ˆz‰‚ ™‘† „
•‚†Žy ~•Œ‡ˆ|Š‚† •‘Œ +∞ { −∞ ~Š‘†•‘Œ|”–•
P
_Š „ ~‡ŒˆŒ’…|~ (sn ) •‚Š z”‚† ™Ž†Œ ‘™‘‚ ˆz‰‚ ™‘† „ •‚†Žy +∞
n=1 xn ~•Œ‡ˆ|Š‚† ‡~† ™‘† •‚Š z”‚†
y…ŽŒ†•‰~
P
m xn ŒŠŒ‰yƒ‚‘~† n Œ•‘™• ™ŽŒ• { •ŽŒ•…‚‘zŒ• ‘„• •‚†Žy• +∞
n=1 xn
P
c•†•„‰~|ŠŒ’‰‚ ™‘† ~Š „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† { ~•Œ‡ˆ|Š‚† •‘Œ ±∞ ‘™‘‚ „ •‚†Žy z”‚†
y…ŽŒ†•‰~ ‡~† ~’‘™ ‚|Š~† ~Ž†…‰™• { ±∞ ~Š‘†•‘Œ|”–• _ŠP„ •‚†Žy ~•Œ‡ˆ|Š‚† ~ˆˆy ™”† •‘~ ±∞
‘™‘‚ „ •‚†Žy •‚Š z”‚† y…ŽŒ†•‰~ nŽŒ•z‹‘‚ ‘Œ •š‰•ŒˆŒ +∞
n=1 xn ‘„• •‚†Žy• ‘–Š xn z”‚† •†•ˆ™
•‚Ž†‚”™‰‚ŠŒ _“ ‚Š™• ‚|Š~† zŠ~ •‡z‘Œ •š‰•ŒˆŒ ~Š‚‹yŽ‘„‘~ ~•™ ‘Œ ~Š „ •‚†Žy z”‚† y…ŽŒ†•‰~
{ ™”† _“ ‚‘zŽŒ’ •‘„Š •‚Ž|•‘–•„ •Œ’ „ •‚†Žy z”‚† y…ŽŒ†•‰~ •’‰•Œˆ|ƒ‚† ‡~† ‘Œ y…ŽŒ†•‰~ ‘„•
•‚†Žy•
P
P+∞
P+∞
qŒ •š‰•ŒˆŒ ‘Œ’ •‚|‡‘„ •‚Š •~|ƒ‚† †•†~|‘‚ŽŒ Ž™ˆŒ ‰‚ +∞
n=1 xn
k=1 xk
j=1 xj •’‰•Œˆ|
ƒŒ’‰‚ ‘„Š |•†~ •‚†Žy
P
n~Žy•‚†€‰~
f‚–ŽŒš‰‚ ‘„ •‚†Žy +∞
n=1 1 { 1+1+1+· · ·+1+· · · q~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘y
‘„• ‚|Š~† Œ† s1 = 1 s2 = 1 + 1 = 2 s3 = 1 + 1 + 1 = 3 ‡~† €‚Š†‡™‘‚Ž~ sn = 1 + · · · + 1 = n
€†~ ‡y…‚ n c•‚†•{ sn = n → +∞ „ •‚†Žy ~•Œ‡ˆ|Š‚† •‘Œ +∞ ‡~† ‘Œ y…ŽŒ†•‰y ‘„• ‚|Š~†
P+∞
n=1 1 = +∞.
P
n~Žy•‚†€‰~
e ~•ˆŒš•‘‚Ž„ •‚†Žy ‚|Š~† „ +∞
e •‚†Žy
n=1 0 { 0 + 0 + 0 + · · · + 0 + · · ·
~’‘{ ŒŠŒ‰yƒ‚‘~† ‰„•‚Š†‡{ •‚†Žy q~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘y ‘„• ‚|Š~† Œ† s1 = 0 s2 = 0 + 0 = 0
s3 = 0 + 0 + 0 = 0 ‡~† €‚Š†‡™‘‚Ž~ sn = 0 + · · · + 0 = 0 €†~ ‡y…‚ n c•‚†•{ sn = 0 → 0 „
‰„•‚Š†‡{ •‚†Žy •’€‡ˆ|Š‚† •‘ŒŠ 0 ‡~† ‘Œ y…ŽŒ†•‰y ‘„• ‚|Š~†
P+∞
n=1 0 = 0.
P
n−1 { 1 + a + a2 + · · · +
n~Žy•‚†€‰~
e €‚–‰‚‘Ž†‡{ •‚†Žy ‰‚ ˆ™€Œ a ‚|Š~† „ +∞
n=1 a
an−1 + · · · e •‚†Žy ~’‘{ z”‚† ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ s1 = 1 s2 = 1 + a s3 = 1 + a + a2 ‡~†
€‚Š†‡™‘‚Ž~ sn = 1 + a + · · · + an−1 €†~ ‡y…‚ n
n~Ž~‘„Ž{•‘‚ ™‘† Œ •Ž›‘Œ• •ŽŒ•…‚‘zŒ• ‘„• €‚–‰‚‘Ž†‡{• •‚†Žy• ‚|Š~† Œ a0 ‡~† ™‘† ‘™Š …‚–
Ž{•~‰‚ |•Œ ‰‚ 1 _’‘™ ‚|Š~† •ŽŒ“~Š›• •–•‘™ ~Š a 6= 0 ~ˆˆy ™”† ~Š a = 0 •†™‘† •‚Š ŒŽ|ƒ‚‘~†
‘Œ •š‰•ŒˆŒ 00 r•yŽ”‚† ™‰–• ‰†~ •~Ž~•Œ•†~‡{ •š‰•~•„ Š~ …‚–Ž‚|‘~† ™‘† a0 = 1 €†~ ‡y…‚P
a
~‡™‰„ ‡~† €†~ a = 0 •‘„Š •‚Ž|•‘–•„ •Œ’ ‚‰“~Š|ƒ‚‘~† ‘Œ •š‰•ŒˆŒ a0 –• ™ŽŒ• •‘Œ •š‰•ŒˆŒ
‘Œ Œ•Œ|Œ ”Ž„•†‰Œ•Œ†Œš‰‚ €†~ Š~ •„ˆ›•Œ’‰‚ •‚•‚Ž~•‰zŠŒ y…ŽŒ†•‰~ P
{ •‚†Žy a†~ •~Žy•‚†€‰~ zŠ~
n
•Œˆ’›Š’‰Œ a0 + a1 x + · · · + aN xN ‰•ŒŽŒš‰‚ Š~ ‘Œ •’‰•Œˆ|•Œ’‰‚ N
n=0 an x ’•ŒŠŒ›Š‘~• ™‘†
x0 = 1 €†~ ‡y…‚ x ~‡™‰„ ‡~† €†~ x = 0
_•™ ‘Œ •~Žy•‚†€‰~
€Š–Ž|ƒŒ’‰‚ •™‘‚ ’•yŽ”‚† ‘Œ y…ŽŒ†•‰~ ‘„• €‚–‰‚‘Ž†‡{• •‚†Žy•
•„ˆ~•{ ‘Œ ™Ž†Œ ‘„• ~‡ŒˆŒ’…|~• (1 + a + · · · + an−1 ) ‡~† ~Š ’•yŽ”‚† ‘„Š ‘†‰{ ‘Œ’ c•Œ‰zŠ–•
€†~ ‘Œ y…ŽŒ†•‰~ ‘„• €‚–‰‚‘Ž†‡{• •‚†Žy• z”Œ’‰‚ ™‘†

= +∞,
~Š a ≥ 1
P+∞ n−1 
= 1/(1 − a), ~Š −1 < a < 1
n=1 a


•‚Š ’•yŽ”‚†, ~Š a ≤ −1
P
n−1 { 1 + (−1) + 1 + (−1) + 1 + (−1) + · · · e •‚†Žy
c†•†‡{ •‚Ž|•‘–•„ ‚|Š~† „ •‚†Žy +∞
n=1 (−1)
~’‘{ •‚Š z”‚† y…ŽŒ†•‰~
P
1
1
1
1
n~Žy•‚†€‰~
e •‚†Žy +∞
n=1 np { 1 + 2p + 3p + · · · + np + · · · ‰‚ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~
1
1
sn = 1 + 2p + · · · + np €†~ ‡y…‚ n
P
1
p‘„Š •‚Ž|•‘–•„ p = 1 •ŽŒ‡š•‘‚† „ •‚†Žy +∞
n=1 n „ Œ•Œ|~ ŒŠŒ‰yƒ‚‘~† ~Ž‰ŒŠ†‡{ •‚†Žy e •‚†Žy
1
~’‘{ z”‚† ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = 1 + 2 + · · · + n1 €†~ ‡y…‚ n ‡~† ‚|•~‰‚ •‘Œ •~Žy•‚†€‰~
™‘† sn → +∞ VŽ~
P+∞ 1
n=1 n = +∞.
P
1
1
c•|•„• •‘„Š •‚Ž|•‘–•„ p = 2 •ŽŒ‡š•‘‚† „ •‚†Žy +∞
n=1 n2 ‰‚ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = 1 + 22 +
· · · + n12 €†~ ‡y…‚ n p‘~ •~Ž~•‚|€‰~‘~
‡~†
‚|•~‰‚ ™‘† „ ~‡ŒˆŒ’…|~ (sn ) •’€‡ˆ|Š‚†
Œ•™‘‚ „ •‚†Žy ~’‘{ •’€‡ˆ|Š‚† ‡~† z”‚† y…ŽŒ†•‰~ ‘Œ Œ•Œ|Œ ‚|Š~† ~Ž†…‰™• b„ˆ~•{
P+∞ 1
e •‚†Žy
n=1 n2 •’€‡ˆ|Š‚†.
P+∞ 1
i|€Œ ~Ž€™‘‚Ž~ …~ ‰‚ˆ‚‘{•Œ’‰‚ ‘„ •‚†Žy n=1 np •‘„ €‚Š†‡{ •‚Ž|•‘–•„
P
1
1
1
1
n~Žy•‚†€‰~
e •‚†Žy +∞
n=1 n! z”‚† ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = 1! + 2! + · · · + n! €†~ ‡y…‚ n
‡~†
™‘† sn → e − 1 Œ•™‘‚ „ •‚†Žy •’€‡ˆ|Š‚† •‘ŒŠ
aŠ–Ž|ƒŒ’‰‚ ~•™ ‘~ •~Ž~•‚|€‰~‘~
~Ž†…‰™ e − 1 ‡~† ‚•Œ‰zŠ–•
P
1
1 + +∞
n=1 n! = e.
P
(−1)n−1
z”‚† ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = 1 − 21 + 31 − 14 + · · · +
n~Žy•‚†€‰~
e •‚†Žy +∞
n=1
n
(−1)n−1
n
™‘† „ ~‡ŒˆŒ’…|~ (sn ) •’€‡ˆ|Š‚† VŽ~ „ •‚†Žy
€†~ ‡y…‚ n c|•~‰‚ •‘Œ •~Žy•‚†€‰~
~’‘{ •’€‡ˆ|Š‚† ‡~† z”‚† y…ŽŒ†•‰~ ‘Œ Œ•Œ|Œ ‚|Š~† ~Ž†…‰™• b„ˆ~•{
P+∞ (−1)n−1
e •‚†Žy
•’€‡ˆ|Š‚†.
n=1
n
‡~† •‘†• ~•‡{•‚†•
‡~†
b‚|‘‚ ‡~† ‘†• ’•Œ•„‰‚†›•‚†• ~’‘›Š ‘–Š ~•‡{•‚–Š
‡~†
nyŠ‘–• „ •‚†Žy ~’‘{ z”‚† ‰‚ˆ‚‘„…‚| ‡~† •‘†• ~•‡{•‚†•
‡~† •‘„Š y•‡„•„
jyˆ†•‘~ •‘„Š y•‡„•„
•Ž{‡~‰‚ ‡~† ‘„Š ‘†‰{ ‘Œ’ ~…ŽŒ|•‰~‘Œ• ‘„• •‚†Žy• ~’‘{•
f~ ‘{Š ‹~Š~•ŽŒš‰‚ •‘Œ •~Žy•‚†€‰~
nomq_pe
_Š „ •‚†Žy
P+∞
n=1 xn
•’€‡ˆ|Š‚† ‘™‘‚ xn → 0
_•™•‚†‹„
f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn
P
x
_Š +∞
n=1 n = s ∈ R ‘™‘‚ sn → s c•‚†•{ †•”š‚† xn = sn − sn−1 €†~ ‡y…‚ n ‰‚ n ≥ 2
•’Š‚•y€‚‘~† xn → s − s = 0
P
n
n
n~Žy•‚†€‰~
e •‚†Žy +∞
n=1 n+1 ~•Œ‡ˆ|Š‚† •†™‘† n+1 → 1 6= 0
P
1
1
n~Žy•‚†€‰~
e ~Ž‰ŒŠ†‡{ •‚†Žy +∞
n=1 n •‚Š •’€‡ˆ|Š‚† [‰–• n → 0
qŒ •~Žy•‚†€‰~ ~’‘™ •‚|”Š‚† ™‘† •‚Š †•”š‚† ‘Œ ~Š‘|•‘ŽŒ“Œ ‘„• •Ž™‘~•„•
P
nyŠ‘–• ™‘~Š …zˆŒ’‰‚ Š~ •Œš‰‚ ~Š •’€‡ˆ|Š‚† ‰†~ •‚†Žy +∞
n=1 xn ‘Œ •Ž›‘Œ •Žy€‰~ •Œ’ ‡y
ŠŒ’‰‚ ‚|Š~† Š~ ‚ˆz€‹Œ’‰‚ ~Š xn → 0 _Š ~’‘™ •‚Š †•”š‚† ‘™‘‚ „ •‚†Žy •‚Š •’€‡ˆ|Š‚† _Š ~’‘™
†•”š‚† ‘™‘‚ •’Š‚”|ƒŒ’‰‚ ‘„Š •ŽŒ••y…‚†y ‰~•
nomq_pe
X•‘–
™‘† Œ† ~‡ŒˆŒ’…|‚• (xn ) ‡~† (yn ) ‘~’‘|ƒŒŠ‘~† ~•™ ‡y•Œ†Œ’• ™ŽŒ’•P‘Œ’• ‡~†
P+∞
•zŽ~ q™‘‚ „ •‚†Žy n=1 xn •’€‡ˆ|Š‚† { ~•Œ‡ˆ|Š‚† •‘Œ ±∞ ~Š ‡~† ‰™ŠŒ ~Š „ •‚†Žy +∞
n=1 yn
~Š‘†•‘Œ|”–• •’€‡ˆ|Š‚† { ~•Œ‡ˆ|Š‚† •‘Œ ±∞
_•™•‚†‹„ X•‘– ™‘† ’•yŽ”Œ’Š k0 , m0 ›•‘‚ Š~ †•”š‚† xk0 +l = ym0 +l €†~ ‡y…‚ l
f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn ‡~† tn = y1 + · · · + yn
a†~ ‡y…‚ l †•”š‚†
sk0 +l − sk0 = xk0 +1 + · · · + xk0 +l = ym0 +1 + · · · + ym0 +l = tm0 +l − tm0 .
VŽ~ Œ† ~‡ŒˆŒ’…|‚• (sn − sk0 ) ‡~† (tn − tm0 ) ‘~’‘|ƒŒŠ‘~† ~•™ ‡y•Œ†Œ’• ™ŽŒ’• ‘Œ’• ‡~† •zŽ~
X•‘–
P+∞
n=1 xn = s ∈ R.
q™‘‚ sn → s Œ•™‘‚ sn −sk0 → s−sk0 VŽ~ tn −tm0 → s−sk0 ‡~† ‚•Œ‰zŠ–• tn → s−sk0 +tm0
VŽ~
P+∞
n=1 yn = s − sk0 + tm0 .
_Š
s∈R
{ s = +∞ {
s = −∞,
‘™‘‚ ~Š‘†•‘Œ|”–•
s − s k 0 + tm 0 ∈ R
n~Žy•‚†€‰~
{ s − sk0 + tm0 = +∞ {
s − sk0 + tm0 = −∞.
X•‘– m ∈ Z j‚ ‘~ •š‰•Œˆ~
P+∞
{
xm + xm+1 + · · · + xm+n−1 + · · ·
n=m xn
•„ˆ›ŠŒ’‰‚ ‘„ •‚†Žy ‰‚ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ t1 = xm t2 = xm + xm+1 ‡~† €‚Š†‡™‘‚Ž~
tn = xm + · · · + xm+n−1
€†~ ‡y…‚ n.
P
P+∞
c|Š~†
‚“~Ž‰™ƒ‚‘~† •‘†• •‚†Žz• +∞
n ‡~†
n=m x
n=1 xn Œ•™‘‚ „ •‚†Žy
P+∞“~Š‚Ž™ ™‘† „ •Ž™‘~•„
P
+∞
x
•’€‡ˆ|Š‚†
{
~•Œ‡ˆ|Š‚†
•‘Œ
±∞
~Š
‡~†
‰™ŠŒ
~Š
„
•‚†Žy
x
n=m n
n=1 n ~Š‘†•‘Œ|”–• •’€‡ˆ|
Š‚† { ~•Œ‡ˆ|Š‚† •‘Œ ±∞ jyˆ†•‘~ ‰•ŒŽŒš‰‚ Š~ •ŽŒš‰‚ ‡~† ‘„ •”z•„ ~Šy‰‚•~ •‘~ ~…ŽŒ|•‰~‘~
‘–Š •šŒ •‚†Ž›Š
p‘„Š •‚Ž|•‘–•„ m ≥ 2 ~Š
s n = x 1 + · · · + xn ,
‘™‘‚ †•”š‚†
tn = xm +· · ·+xm+n−1 = (x1 +· · ·+xm+n−1 )−(x1 +· · ·+xm−1 ) = sm+n−1 −(x1 +· · ·+xm−1 )
€†~ ‡y…‚ n _Š
P+∞
n=1 xn
= s ∈ R,
‘™‘‚ sn → s Œ•™‘‚ tn → s − (x1 + · · · + xm−1 ) ‡~† ‚•Œ‰zŠ–•
P+∞
n=m xn = s − (x1 + · · · + xm−1 ).
VŽ~
P+∞
n=1 xn
= x1 + · · · + xm−1 +
p‘„Š •‚Ž|•‘–•„ m ≤ 0 †•”š‚†
P+∞
n=m xn
~Š m ≥ 2.
tn = xm + · · · + xm+n−1 = (xm + · · · + x0 ) + (x1 + · · · + xm+n−1 ) = (xm + · · · + x0 ) + sm+n−1
€†~ ‡y…‚ n ≥ 2 − m _Š
P+∞
n=1 xn
= s ∈ R,
‘™‘‚ sn → s Œ•™‘‚ tn → xm + · · · + x0 + s ‡~† ‚•Œ‰zŠ–•
P+∞
n=m xn = xm + · · · + x0 + s.
VŽ~
P+∞
n=m xn
= xm + · · · + x0 +
P+∞
n=1 xn
~Š m ≤ 0.
p’Š•’yƒŒŠ‘~• ‘Œ’• ‘š•Œ’•
‡~†
‚š‡Œˆ~ •ˆz•Œ’‰‚ ™‘† ~Š m, k ∈ Z ‡~† m < k ‘™‘‚
P+∞
P+∞
~Š m < k.
n=m xn = xm + · · · + xk−1 +
n=k xn
p‘ŒŠ ”‚†Ž†•‰™ ‘–ŠP
•‚†Ž›Š ‚‰“~Š|ƒ‚‘~† ‰‚Ž†‡z• “ŒŽz• ‰†~ ~•ˆ{ ~ˆˆ~€{ ‰‚‘~•ˆ„‘{• a†~ •~Žy
•‚†€‰~ z•‘– „ •‚†Žy +∞
n=m xn c†•y€Œ’‰‚ ‘„ Šz~ ‰‚‘~•ˆ„‘{ k = n − m + 1 ‡~† •ˆz•Œ’‰‚ ™‘†
™‘~Š Œ n •†~‘Žz”‚† ‘Œ’• m, m + 1, m + 2, . . . ‘™‘‚ Œ k •†~‘Žz”‚† ‘Œ’• 1, 2, 3, . . . VŽ~
P+∞
P+∞
n=m xn =
k=1 xk+m−1 .
nŽy€‰~‘† ‡~† Œ† •šŒ •‚†Žz• ‚|Š~† „ xm + xm+1 + xm+2 + · · ·
P
P+∞
nomq_pe
X•‘– ™‘† „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ‡~† z•‘– rn =
m=n xm €†~ ‡y…‚ n q™‘‚
rn → 0
P
_•™•‚†‹„ f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn _Š +∞
n=1 xn = s ∈ R ‘™‘‚
sn → s c•‚†•{ ~•™ ‘„Š
{ ‡~† ‘„Š €‚Š†‡™‘‚Ž„
z”Œ’‰‚ ™‘† †•”š‚†
s = sn−1 + rn
€†~ ‡y…‚ n •’Š‚•y€‚‘~† rn = s − sn−1 → s − s = 0
P
P+∞
P+∞
P+∞
nomq_pe
_Š Œ† •‚†Žz• +∞
n z”Œ’Š y…ŽŒ†•‰~ ‡~† ‘Œ
n=1 xn ‡~†
n=1
n=1 xn +
n=1 yn
Py+∞
•‚Š ‚|Š~† ~•ŽŒ••†™Ž†•‘„ ‰ŒŽ“{ ‘™‘‚ ‡~† „ •‚†Žy n=1 (xn + yn ) z”‚† y…ŽŒ†•‰~ ‡~†
P+∞
P+∞
P+∞
n=1 (xn + yn ) =
n=1 xn +
n=1 yn .
P
P+∞
_•™•‚†‹„ X•‘– +∞
n=1 xn = s ∈ R ‡~†
n=1 yn = t ∈ R f‚–ŽŒš‰‚ ‡~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~
sn = x1 + · · · + xn ‡~† tn = y1 +P
· · · + yn
q~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚†Žy• +∞
n=1 (xn + yn ) ‚|Š~† ‘~
un = (x1 + y1 ) + · · · + (xn + yn ) = (x1 + · · · + xn ) + (y1 + · · · + yn ) = sn + tn .
c•Œ‰zŠ–•
un = sn + tn → s + t.
P+∞
P
VŽ~ n=1 (xn + yn ) = s + t = n=1 xn + +∞
n=1 yn
P+∞
P+∞
nomq_pe
_Š „ •‚†Žy
P+∞ n=1 xn z”‚† y…ŽŒ†•‰~ ‡~† ‘Œ λ n=1 xn •‚Š ‚|Š~† ~•ŽŒ••†™Ž†•‘„
‰ŒŽ“{ ‘™‘‚ ‡~† „ •‚†Žy n=1 λxn z”‚† y…ŽŒ†•‰~ ‡~†
P+∞
P+∞
n=1 λxn = λ
n=1 xn .
P
_•™•‚†‹„ X•‘– +∞
n=1 xn = s ∈ R
P f‚–ŽŒš‰‚ ‡~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn
q~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚†Žy• +∞
n=1 λxn ‚|Š~† ‘~
P+∞
wn = λx1 + · · · + λxn = λ(x1 + · · · + xn ) = λsn .
VŽ~
‡~† ‚•Œ‰zŠ–•
wn = λsn → λs
P+∞
n=1 λxn = λs = λ
n=1 xn
P+∞
j•ŒŽŒš‰‚ Š~ •’Š•’y•Œ’‰‚ ‘~ •šŒ ‘‚ˆ‚’‘~|~ ~•Œ‘‚ˆz•‰~‘~ –• ‚‹{•
P+∞
P+∞
P+∞
n=1 (λxn + µyn ) = λ
n=1 xn + µ
n=1 yn .
cš‡Œˆ~ •ˆz•Œ’‰‚ ‰‚ ‚•~€–€{ ™‘† ~’‘™ †•”š‚† €†~ Œ•Œ†‚••{•Œ‘‚ •‚•‚Ž~•‰zŠŒ’ •ˆ{…Œ’• •‚†Žz•
nomq_pe
X•‘– ™‘† †•”š‚† xn ≤ yn €†~ ‡y…‚ n
P+∞
P
P+∞
P+∞
>~@ _Š Œ† •‚†Žz• n=1 xn ‡~† +∞
yn
n=1 yn z”Œ’Š y…ŽŒ†•‰~ ‘™‘‚
n=1 xn ≤
P+∞
P+∞n=1
_Š
‚•†•ˆzŒŠ P
’•yŽ”‚† n0 ›•‘‚ xn0 < yP
n0 ‡~† ~Š Œ† •‚†Žz•
P+∞
P+∞ n=1 xn ‡~† n=1 yn •’€‡ˆ|ŠŒ’Š ‘™‘‚
+∞
+∞
x
<
y
g•Œ•šŠ~‰~
~Š
x
=
n=1 n
n=1 n
n=1 n
n=1 yn ‡~† ‘Œ ‡Œ†Š™ y…ŽŒ†•‰~ ‚|Š~† ~Ž†…‰™•
‘™‘‚ xn = yn €†~ ‡y…‚ n
P
P+∞
>•@ _Š +∞
n=1 xn = +∞ ‘™‘‚
n=1 yn = +∞
P+∞
P+∞
>€@ _Š n=1 yn = −∞ ‘™‘‚ n=1 xn = −∞
P
P+∞
_•™•‚†‹„ >~@ X•‘– +∞
n=1 xn = s ∈ R ‡~†
n=1 yn = t ∈ R
f‚–ŽŒš‰‚ ‡~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn ‡~† tn = y1 + · · · + yn c•‚†•{ sn → s
‡~† tn → t ‡~† ‚•‚†•{ †•”š‚†
s n = x1 + · · · + xn ≤ y 1 + · · · + y n = t n
€†~ ‡y…‚ n •’Š‚•y€‚‘~† s ≤ t
X•‘– ‚•†•ˆzŒŠ xn0 < yn0 ‡~† s, t ∈ R q™‘‚ €†~ ‡y…‚ n ≥ n0 †•”š‚†
tn − sn = (yn − xn ) + · · · + (y1 − x1 ) ≥ yn0 − xn0 ,
‚•‚†•{ ™ˆ‚• Œ† •~Ž‚Š…z•‚†• ‚|Š~† ‰„ ~ŽŠ„‘†‡z• ‡~† Œ yn0 − xn0 ‚|Š~† zŠ~• ~•™ ‘Œ’• ™ŽŒ’• ‘Œ’
~…ŽŒ|•‰~‘Œ• VŽ~ t − s ≥ yn0 − xn0 > 0
>•@ [•–• •Ž†Š †•”š‚† sn ≤ tn €†~ ‡y…‚ n VŽ~ ~Š sn → +∞ ‘™‘‚ tn → +∞
>€@ [•–• •‘Œ >•@
_•‡{•‚†•
P+∞ P+∞ 1
P
1 P+∞
n
√ −√ 1
`Ž‚|‘‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘–Š +∞
n=1 (−1)
n=1 n
n=1
n=1 (− 2 )
n
n+1
P+∞ 2 P+∞
P+∞
n−1 n
n−1 n2 ‡~† •Ž‚|‘‚ ‘~ ~…ŽŒ|•‰~‘y ‘Œ’• ~Š ’•yŽ”Œ’Š
n=1 n
n=1 (−1)
n=1 (−1)
P+∞
P+∞ n n P+∞ √
P
1
n
n
n
c‹‚‘y•‘‚ –• •ŽŒ• ‘„ •š€‡ˆ†•„ ‘†• +∞
n=1 n VLQ n
n=1 ( n+1 )
n=1
n=1 2n+1
P
P+∞
n
n
+∞
2 +3
1
n=1 2n+1 +3n+1
n=1 n ORJ(1 + n )
P+∞ 2 n+2
( )
tŽ„•†‰Œ•Œ†›Š‘~• €‚–‰‚‘Ž†‡z• •‚†Žz• ‚‹‚‘y•‘‚ –• •ŽŒ• ‘„ •š€‡ˆ†•„ ‘†• n=1
P+∞ 4 n−3 P+∞
P+∞ 2 n P+∞
P+∞ 2 P+∞ 2n−1 +3n+13−6n/2
n−4
n
n=1
n=1 ( 3 )
n=3 (− 3 )
n=1 3n−1
n=1 (−1)
n=4 (−3)
6n
‡~† •Ž‚|‘‚ ‘~ ~…ŽŒ|•‰~‘y ‘Œ’• ~Š ’•yŽ”Œ’Š
P
P+∞
P+∞ 1−x2n
x
x2n
a†~ •Œ†z• ‘†‰z• ‘Œ’ x •’€‡ˆ|ŠŒ’Š Œ† •‚†Žz• +∞
n=1 (1+x)n−1
n=1 (1+x2 )n−1
n=1 1+x2n
P
`Ž‚|‘‚ •’ŠŒ•‘†‡™ ‘š•Œ €†~ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚†Žy• +∞
n=1 (bn − bn+1 ) ‡~† •y
•‚† ~’‘Œš ~•Œ•‚|‹‘‚ ™‘† ~’‘{ z”‚† y…ŽŒ†•‰~ ~Š ‡~† ‰™ŠŒ ~Š ’•yŽ”‚† ‘Œ OLPn→+∞ bn ‡~† ™‘† ‘Œ
y…ŽŒ†•‰~ ‚|Š~† ~Ž†…‰™• ~Š ‡~† ‰™ŠŒ ~Š ‘Œ OLPn→+∞ bn ‚|Š~† ~Ž†…‰™• _•Œ•‚|‹‘‚ ‘„Š ‚‹{• •”z•„
~Šy‰‚•~ •‘Œ y…ŽŒ†•‰~ ‘„• •‚†Žy• ‡~† •‘Œ OLPn→+∞ bn
P+∞
n=1 (bn − bn+1 ) = b1 − OLPn→+∞ bn .
P+∞ 1
P+∞
P+∞
1
n P+∞
1
`Ž‚|‘‚ ‘~ ~…ŽŒ|•‰~‘~ ‘–Š n=1 n(n+1)
n=1 (2n−1)(2n+1)
n=1 ORJ n+1
n=1 n(n+1)(n+2)
√
√
P+∞ n+1− n P+∞
n−1 2n+1
√
n=1
n=1 (−1)
n(n+1)
n2 +n
P
1
1
`Ž‚|‘‚ ~Š ’•yŽ”‚† ‘Œ y…ŽŒ†•‰~ ‘„• •‚†Žy• +∞
n=1 xn ™•Œ’ x2k−1 = k ‡~† x2k = − k €†~
‡y…‚ k
P+∞
P+∞
X•‘– x 6= y _Š
Œ†
•‚†Žz•
(a
+
xa
)
‡~†
2k
2k−1
k=1
k=1 (a2k + ya2k−1 ) •’€‡ˆ|ŠŒ’Š
P+∞
~•Œ•‚|‹‘‚ ™‘† „ •‚†Žy n=1 an •’€‡ˆ|Š‚†
(
P+∞ x2n−1
x/(1 − x), ~Š |x| < 1
_•Œ•‚|‹‘‚ ™‘† n=1 1−x2n =
1/(1 − x), ~Š |x| > 1
P+∞ 2n−1
1
_Š x > 1 ~•Œ•‚|‹‘‚ ™‘† n=1 2n−1 = x−1
x
+1
(
1
P
~Š n 6= k
2
2,
`Ž‚|‘‚ ‘Œ y…ŽŒ†•‰~ ‘„• •‚†Žy• +∞
X•‘– k ∈ N ‡~† xn = n −k
n=1 xn
0,
~Š n = k
j†~ ‚•|•‚•„ Š†“y•~ ”†ŒŠ†Œš ’“|•‘~‘~† •†~•Œ”†‡z• ~ˆˆ~€z• ‰‚ ‘ŒŠ ‚‹{• ‘Ž™•Œ qŒ ~Ž”†‡™
•”{‰~ ‘„• ‚|Š~† †•™•ˆ‚’ŽŒ ‘Ž|€–ŠŒ •ˆ‚’Žy• ‰{‡Œ’• s h~‘™•†Š ~•™ ‘Œ ‰‚•~|Œ zŠ~ ‘Ž|‘Œ ‡y…‚
•ˆ‚’Žy• ‹‚“’‘Ž›Š‚† zŠ~ †•™•ˆ‚’ŽŒ ‘Ž|€–ŠŒ Œ•™‘‚ ‘Œ ŠzŒ •”{‰~ ‘„• Š†“y•~• ‚|Š~† •Œˆ’€–Š†‡™
‰‚ 12 †•Œ‰{‡‚†• •ˆ‚’Žz• h~‘™•†Š ~•™ ‘Œ ‰‚•~|Œ zŠ~ ‘Ž|‘Œ ‡y…‚ •ˆ‚’Žy• ‘„• Šz~• Š†“y•~• ‹‚“’
‘Ž›Š‚† zŠ~ †•™•ˆ‚’ŽŒ ‘Ž|€–ŠŒ Œ•™‘‚ ‘Œ ŠzŒ •”{‰~ ‘„• Š†“y•~• ‚|Š~† •Œˆ’€–Š†‡™ ‰‚ 48 †•Œ‰{‡‚†•
•ˆ‚’Žz• _Š ~’‘{ „ •†~•†‡~•|~ •’Š‚”†•‘‚| ‚• y•‚†ŽŒŠ “~Š‘~•‘‚|‘‚ ‘Œ ŒŽ†~‡™ •”{‰~ ‘„• Š†“y•~•
‡~† ’•ŒˆŒ€|•‘‚ ‘Œ ‰{‡Œ• ‘„• •‚Ž†“zŽ‚†~• ‡~† ‘Œ ‚‰•~•™ ‘„• ŒŽ†~‡{• Š†“y•~•
XŠ~ ~’‘Œ‡|Š„‘Œ ‹‚‡†Šy ~•™ ‘„Š •™ˆ„ _ ‡~† •‚ ‚’…š •Ž™‰Œ ‡~‘‚’…šŠ‚‘~† •ŽŒ• ‘„Š •™ˆ„
` ‰‚ •‘~…‚Ž{ ‘~”š‘„‘~ v [ˆŒ† €Š–Ž|ƒŒ’‰‚ ™‘† ~Š „ ~•™•‘~•„ ‘–Š •šŒ •™ˆ‚–Š ‚|Š~† d ‘™‘‚ ‘Œ
~’‘Œ‡|Š„‘Œ …~ ŒˆŒ‡ˆ„Ž›•‚† ‘„ •†~•ŽŒ‰{ •‚ ”Ž™ŠŒ vd _•~Š‘{•‘‚ ™‰–• •‚ ‡y•Œ†ŒŠ •Œ’ †•”’Ž|
ƒ‚‘~† ™‘† ‘Œ ~’‘Œ‡|Š„‘Œ •‚Š …~ “‘y•‚† •Œ‘z •‘„Š •™ˆ„ ` ‡~† ‘Œ •†‡~†ŒˆŒ€‚| –• ‚‹{•
_• ’•Œ…z•Œ’‰‚ ™‘† ‘Œ ~’‘Œ‡|Š„‘Œ ‡~ˆš•‘‚† ‘„ ‰†•{ ~•™•‘~•„ ‡~† ‰yˆ†•‘~ •‘ŒŠ •ŽŒ•ˆ‚•™‰‚ŠŒ
€† ~’‘{ ”Ž™ŠŒ _• ’•Œ…z•Œ’‰‚ ‚•|•„• ™‘† ‡~‘™•†Š ‘Œ ~’‘Œ‡|Š„‘Œ ‡~ˆš•‘‚† ‘„ ‰†•{ ~•™ ‘„Š ‚Š~
•Œ‰zŠŒ’•~ ~•™•‘~•„ •‘ŒŠ •ŽŒ•ˆ‚•™‰‚ŠŒ €† ~’‘{ ”Ž™ŠŒ h~† Œš‘– ‡~… ‚‹{• qŒ ~’‘Œ‡|Š„‘Œ z”‚†
hy…‚ •‚†Žy ‘„• ‰ŒŽ“{•
P+∞
n=1 (bn
− bn+1 ) ”~Ž~‡‘„Ž|ƒ‚‘~† ‘„ˆ‚•‡Œ•†‡{ •‚†Žy
™‰–• •yŠ‘Œ‘‚ ‰•ŽŒ•‘y ‘Œ’ ‡y•Œ†~ ‚Š~•Œ‰zŠŒ’•~ z•‘– ‡~† •Œˆš ‰†‡Ž{ ~•™•‘~•„ ‰z”Ž† ‘„Š •™ˆ„
` Œ•™‘‚ •‚Š …~ “‘y•‚† •Œ‘z ‚‡‚|
e ~•yŠ‘„•{ •~• €†~ Š~ ‚|Š~† •‚†•‘†‡{ •Žz•‚† Œ•–••{•Œ‘‚ Š~ ~‡ŒˆŒ’…{•‚† ‘~ ˆŒ€†‡y •{‰~‘~ ‘Œ’
•~Ž~•yŠ– †•”’Ž†•‰Œš
p‚ ‡y…‚ •‚†Žy ~Š‘†•‘Œ†”‚| „ ~‡ŒˆŒ’…|~ ‘–Š ‰‚Ž†‡›Š ~…ŽŒ†•‰y‘–Š ‘„• _•Œ•‚|‹‘‚ ™‘† ~Š‘†
•‘Ž™“–• •‚ ‡y…‚ ~‡ŒˆŒ’…|~ ~Š‘†•‘Œ†”‚| ‰†~ •‚†Žy z‘•† ›•‘‚ „ ~‡ŒˆŒ’…|~ ~’‘{ Š~ ‘~’‘|ƒ‚‘~† ‰‚
‘„Š ~‡ŒˆŒ’…|~ ‘–Š ‰‚Ž†‡›Š ~…ŽŒ†•‰y‘–Š ‘„• •‚†Žy•
p‚†Žz• ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’•
qŒ …‚›Ž„‰~
z”‚† €†~ ‘†• •‚†Žz• ‘ŒŠ |•†Œ Ž™ˆŒ •Œ’ z”‚† €†~ ‘†• ~‡ŒˆŒ’…|‚• ‘Œ …‚›Ž„‰~
[•–• ‘Œ …‚›Ž„‰~
‚‹~•“~ˆ|ƒ‚† ™‘† Œ† ‰ŒŠ™‘ŒŠ‚• ~‡ŒˆŒ’…|‚• z”Œ’Š Œ•–••{•Œ‘‚ ™Ž†Œ ‚Š› „
‘’”Œš•~ ~‡ŒˆŒ’…|~ ‰•ŒŽ‚| Š~ ‰„Š z”‚† ™Ž†Œ z‘•† ‡~† ‘Œ …‚›Ž„‰~
‚‹~•“~ˆ|ƒ‚† ™‘† Œ† •‚†Žz•
‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• z”Œ’Š Œ•–••{•Œ‘‚ y…ŽŒ†•‰~ ‚Š› „ ‘’”Œš•~ •‚†Žy ‰•ŒŽ‚| Š~ ‰„Š z”‚†
y…ŽŒ†•‰~ jyˆ†•‘~ ™•–• …~ “~Š‚| ~‰z•–• „ ~•™•‚†‹„ ‘Œ’ …‚–Ž{‰~‘Œ•
”Ž„•†‰Œ•Œ†‚| ‘Œ
…‚›Ž„‰~
P
fcvoej_
_Š †•”š‚† xn ≥ 0P€†~ ‡y…‚ n ‘™‘‚ „ •‚†Žy +∞
n=1 xn z”‚† y…ŽŒ†•‰~ ‡~† ~’‘™ ‚|Š~†
+∞
~Ž†…‰™• ≥ 0 { +∞ b„ˆ~•{ 0 ≤ n=1 xn ≤ +∞
n†Œ •’€‡‚‡Ž†‰zŠ~ z•‘– sn = x1 +· · ·+xn ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚†Žy• q™‘‚ „ •‚†Žy •’€‡ˆ|Š‚†
~Š „ ~‡ŒˆŒ’…|~ (sn ) ‚|Š~† yŠ– “Ž~€‰zŠ„ ‡~† ~•Œ‡ˆ|Š‚† •‘Œ +∞ ~Š „ (sn ) •‚Š ‚|Š~† yŠ– “Ž~€‰zŠ„
_•™•‚†‹„ g•”š‚†
sn+1 = x1 + · · · + xn + xn+1 = sn + xn+1 ≥ sn
€†~ ‡y…‚ n VŽ~ „ (sn ) ‚|Š~† ~š‹Œ’•~
‡~† ‚•Œ‰zŠ–• z”‚† ™Ž†Œ ‘Œ Œ•Œ|Œ ‚|Š~† ~Ž†…‰™• { +∞
P+∞
X•‘– sn → s ∈ R ∪ {+∞} q™‘‚ n=1 xn = s
c•‚†•{ †•”š‚†
s n = x1 + · · · + x n ≥ 0
€†~ ‡y…‚ n •’Š‚•y€‚‘~† s ≥ 0
_Š „ (sn ) ‚|Š~† yŠ– “Ž~€‰zŠ„ •’Š‚•y€‚‘~† s ∈ R ‚Š› ~Š „ (sn ) •‚Š ‚|Š~† yŠ– “Ž~€‰zŠ„
•’Š‚•y€‚‘~† s = +∞
P
VŽ~ ‡y…‚ •‚†Žy +∞
n=1 xn ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’•
P+∞z”‚† y…ŽŒ†•‰~ ‡~† ‘Œ y…ŽŒ†•‰~ ~’‘™ ‚|
Š~† ~Ž†…‰™• ≥ P
0 { +∞ c•|•„• ‘Œ ™‘† ‰†~ •‚†Žy n=1 xn ‰‚ ‰„ P
~ŽŠ„‘†‡Œš• ™ŽŒ’• •’€‡ˆ|Š‚†
+∞
†•Œ•’Š~‰‚| ‰‚ n=1
xn < +∞ ‡~† ‘Œ ™‘† ~•Œ‡ˆ|Š‚† †•Œ•’Š~‰‚| ‰‚ +∞
n=1 xn = +∞
P
P+∞
nomq_pe
>~@ X•‘–
q™‘‚ 0 ≤ +∞
n=1 xn ≤
n=1 yn
P+∞™‘† †•”š‚† 0 ≤ xn ≤ yn €†~ ‡y…‚Pn+∞
_Š ‚•†•ˆzŒŠ „ •‚†Žy n=1 yn •’€‡ˆ|Š‚† ‘™‘‚ ‡~† „ •‚†Žy n=1 xn •’€‡ˆ|Š‚†
>•@ X•‘– ™‘† †•”š‚† xn ≥ 0 ‡~† yn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† „ ~‡ŒˆŒ’…|~ ( xynn ) •’€‡ˆ|Š‚† {
P
P+∞
€‚Š†‡™‘‚Ž~ ‚|Š~† “Ž~€‰zŠ„ _Š „ •‚†Žy +∞
n=1 yn •’€‡ˆ|Š‚† ‘™‘‚ ‡~† „ •‚†Žy
n=1 xn •’€‡ˆ|Š‚†
P+∞
P+∞
_•™•‚†‹„ >~@ pš‰“–Š~ ‰‚ ‘Œ …‚›Ž„‰~
Œ† •‚†Žz• n=1 xn ‡~† n=1 yn z”Œ’Š y…ŽŒ†•‰~
•’Š‚•y€‚‘~†
Œ•™‘‚ ~•™ ‘„Š •Ž™‘~•„
P
P+∞
0 ≤ +∞
n=1 xn ≤
n=1 yn .
P
P+∞
P+∞
z”Œ’‰‚ +∞
_Š „ •‚†Žy n=1 yn •’€‡ˆ|Š‚†
n=1 xn < +∞
P+∞ ‘™‘‚ n=1 yn < +∞ _•™ ‘„Š
‡~† ‚•Œ‰zŠ–• „ •‚†Žy n=1 xn •’€‡ˆ|Š‚†
>•@ _Š „ ~‡ŒˆŒ’…|~ ( xynn ) ‚|Š~† “Ž~€‰zŠ„ ’•yŽ”‚† u ›•‘‚ Š~ †•”š‚† xynn ≤ u ‡~† ‚•Œ‰zŠ–•
P+∞
xn ≤ uyn
€†~ ‡y…‚ n c•‚†•{ „ •‚†Žy n=1 yn •’€‡ˆ|Š‚† P
•’Š‚•y€‚‘~† ™‘† ‡~† „ •‚†Žy
VŽ~ •š‰“–Š~ ‰‚ ‘„Š
‡~† ‘Œ >~@ „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚†
P+∞
n=1 uyn
•’€‡ˆ|Š‚†
P
qŒ ~•Œ‘zˆ‚•‰~ ‘„• •Ž™‘~•„• P
>•@ •†~‘’•›Š‚‘~† †•Œ•šŠ~‰~ –• ‚‹{• ~Š „ •‚†Žy +∞
n=1 xn
+∞
~•Œ‡ˆ|Š‚† •‘Œ +∞ ‘™‘‚ ‡~† „ •‚†Žy n=1 yn ~•Œ‡ˆ|Š‚† •‘Œ +∞
_Ž‡‚‘z• “ŒŽz• ‚“~Ž‰™ƒŒ’‰‚ ‘„Š •Ž™‘~•„ >•@ ‰‚ ‘ŒŠ ‚‹{• ‘Ž™•Œ _Š †•”š‚† xn , yn > 0 €†~
‡y…‚ n ‡~† xynn → ρ ™•Œ’ Œ ρ ‚|Š~† zŠ~• …‚‘†‡™• ~Ž†…‰™• •„ˆ~•{ 0 < ρ < +∞ ‘™‘‚ ‘Œ •’‰•zŽ~
P
P+∞
•‰~ €†~ ‘†• •šŒ •‚†Žz• +∞
n=1 xn ‡~†
n=1 yn ‚|Š~† ‘Œ ‚‹{• ‚|‘‚ ‡~† Œ† •šŒ •‚†Žz• •’€‡ˆ|ŠŒ’Š ‚|‘‚
‡~† Œ† •šŒ ~•Œ‡ˆ|ŠŒ’Š •‘Œ +∞ nŽy€‰~‘† ~•™ ‘Œ xynn → ρ ‡~† ‘Œ ™‘† Œ ρ ‚|Š~† ~Ž†…‰™• •’Š‚•y
P
P+∞
€‚‘~† ™‘† ~Š „ •‚†Žy +∞
•’€‡ˆ|Š‚† _ˆˆy ‡~† ~•™ ‘Œ
n=1 yn •’€‡ˆ|Š‚† ‘™‘‚ ‡~† „ •‚†Žy
n=1 x
Pn+∞
yn
1
1
‡~†
‘Œ
™‘†
Œ
‚|Š~†
~Ž†…‰™•
•’Š‚•y€‚‘~†
™‘†
~Š
„
•‚†Žy
→
n=1 xn •’€‡ˆ|Š‚† ‘™‘‚ ‡~† „
xn
ρ
ρ
P
+∞
•‚†Žy n=1 yn •’€‡ˆ|Š‚†
_Š †•”š‚† xn , yn > 0 €†~ ‡y…‚ n ‡~† xynn → 0 •’‰•‚Ž~|ŠŒ’‰‚ •š‰“–Š~ •yŠ‘~ ‰‚ ‘„Š •Ž™‘~•„
P+∞
P
>•@ ™‘† ~Š „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† qŒ ~Š‘|•‘ŽŒ“Œ
n=1 yn •’€‡ˆ|Š‚† ‘™‘‚ ‡~† „ •‚†Žy
™‰–• •‚Š †•”š‚† €‚Š†‡y
P+∞ 1
2
P+∞ 1
n~Žy•‚†€‰~
g•”š‚† 1/n
n=1 n2 < +∞ ‡~†
n=1 n = +∞
1/n → 0
P
`z•~†~ ~Š †•”š‚† xn , yn > 0 €†~ ‡y…‚ n ‡~† xynn → +∞ •’‰•‚Ž~|ŠŒ’‰‚ ™‘† ~Š „ •‚†Žy +∞
n=1 xn
P+∞
yn
•’€‡ˆ|Š‚† ‘™‘‚ ‡~† „ •‚†Žy n=1 yn •’€‡ˆ|Š‚† h~† •yˆ† ~’‘™ •ŽŒ‡š•‘‚† ~•™ ‘Œ xn → 0 ‡~† ‘„Š
•Ž™‘~•„ >•@
j†~ ‘‚ˆ‚’‘~|~ •~Ž~‘{Ž„•„ z”‚† Š~ ‡yŠ‚† ‰‚ ‘ŒŠ ‘Ž™•Œ •Œ’ ‚“~Ž‰™ƒ‚‘~† •Œˆˆz• “ŒŽz• „ •š
€‡Ž†•„ •‚†Ž›Š •‘Œ •ˆ~|•†Œ ‘„• •Ž™‘~•„•
_Š z”Œ’‰‚ ‰†~ •‚†Žy ‰‚ •‚Ž|•ˆŒ‡Œ’• •ŽŒ•…‚‘zŒ’•
•ŽŒ••~…Œš‰‚ Š~ ‘„ •’€‡Ž|ŠŒ’‰‚ ‰‚ ‰†~ •‚†Žy ‰‚ ~•ˆŒš•‘‚ŽŒ’• •ŽŒ•…‚‘zŒ’• ›•‘‚ Š~ ‚|Š~† •†Œ
‚š‡ŒˆŒ Š~ ~•Œ“~Š…Œš‰‚ €†~ ‘„ •š€‡ˆ†•„ { ~•™‡ˆ†•„ ~’‘{• ‘„• ~•ˆŒš•‘‚Ž„• •‚†Žy• e ‰‚‘y•~•„
~•™ ‘Œ’• •‚Ž|•ˆŒ‡Œ’• •‘Œ’• ~•ˆŒš•‘‚ŽŒ’• •ŽŒ•…‚‘zŒ’• €|Š‚‘~† •Œˆˆz• “ŒŽz• ‰‚ ‘„Š ~Š~€Š›Ž†•„
‡šŽ†–Š ™Ž–Š ™•–• …~ “~Š‚| •‘~ ‚•™‰‚Š~ •~Ž~•‚|€‰~‘~ j‚€yˆ„ •Œ{…‚†~ •‘„Š ~Š~€Š›Ž†•„ ‡š
Ž†–Š ™Ž–Š •~Žz”‚† „ ‡~ˆ{ ‡~‘~Š™„•„ ‘„• †‚ŽyŽ”„•„• ‘y‹‚–Š ‰‚€z…Œ’• ‡~…›• ‡~† „ ‡~‘~Š™„•„
•†~“™Ž–Š ŒŽ†~‡›Š •’‰•‚Ž†“ŒŽ›Š
P
2n +3
n~Žy•‚†€‰~
f‚–ŽŒš‰‚ ‘„ •‚†Žy +∞
n=1 3n−1 +n
n
2 +3
‚|Š~† •ŽŒ“~Š›• Œ† 2n ‡~† 3n−1
m† ‡šŽ†Œ† ™ŽŒ† •‘ŒŠ ~Ž†…‰„‘{ ‡~† ‘ŒŠ •~ŽŒŠŒ‰~•‘{ ‘Œ’ 3n−1
+n
~Š‘†•‘Œ|”–• m•™‘‚ €Žy“Œ’‰‚
2n +3
1+3·2−n
2n
= 3n−1
3n−1 +n
1+n3−n+1
€†~ ‡y…‚ n ‡~‘™•†Š •ˆz•Œ’‰‚ ™‘†
1+3·2−n
1+n3−n+1
→ 1 ‡~† ‚•Œ‰zŠ–•
n
n
2
2 +3
)/( 3n−1
) → 1.
( 3n−1
+n
P
P+∞ 2 n−1
2n
q›Ž~ •’€‡Ž|ŠŒ’‰‚ ‘„Š ~Ž”†‡{ •‚†Žy ‰‚ ‘„ •‚†Žy +∞
c•‚†•{ ~’‘{ „
n=1 3n−1 = 2
n=1 ( 3 )
‘‚ˆ‚’‘~|~ •‚†Žy •’€‡ˆ|Š‚† •’Š‚•y€‚‘~† ™‘† ‡~† „ ~Ž”†‡{ •‚†Žy •’€‡ˆ|Š‚†
P
2n3 +3n+5
n~Žy•‚†€‰~
X•‘– „ •‚†Žy +∞
n=1 n4 +5n2 +n+7
q›Ž~ Œ† ‡šŽ†Œ† ™ŽŒ† •‘ŒŠ ~Ž†…‰„‘{ ‡~† •‘ŒŠ •~ŽŒŠŒ‰~•‘{ ‘Œ’
~Š‘†•‘Œ|”–• q™‘‚ €Žy“Œ’‰‚
2n3 +3n+5
n4 +5n2 +n+7
‡~‘™•†Š •ˆz•Œ’‰‚ ™‘†
=
1+(3/2)n−2 +(5/2)n−3
1+5n−2 +n−3 +7n−4
3
2n3 +3n+5
n4 +5n2 +n+7
‚|Š~† Œ† 2n3 ‡~† n4
2n3 1+(3/2)n−2 +(5/2)n−3
,
n4 1+5n−2 +n−3 +7n−4
→ 1 ‡~† ‚•Œ‰zŠ–•
3
+3n+5
)/( 2n
) → 1.
( n42n
+5n2 +n+7
n4
c•‚†•{ 1 > 0 ‡~†
2n3
n=1 n4
P+∞
=2
P+∞
1
n=1 n
= +∞ •’Š‚•y€‚‘~†
2n3 +3n+5
n=1 n4 +5n2 +n+7
P+∞
= +∞
n~Žy•‚†€‰~
X•‘– „ •‚†Žy
aŠ–Ž|ƒŒ’‰‚ ‘Œ ™Ž†Œ
P+∞
1
n=1 VLQ n
VLQ x
x
= 1.
P
VŽ~ …~ •’€‡Ž|ŠŒ’‰‚ ‘„ •‚†Žy •Œ’ z”Œ’‰‚ ‰‚ ‘„ •‚†Žy +∞
n=1
OLPx→0
VLQ(1/n)
1/n
1
n
c•‚†•{
→ 1,
P
1
= +∞ •’Š‚•y€‚‘~† +∞
n=1 VLQ n = +∞
P
n3
n~Žy•‚†€‰~
X•‘– „ •‚†Žy +∞
n=1 2n
f‚–ŽŒš‰‚ zŠ~Š Œ•Œ†ŒŠ•{•Œ‘‚ a ‰‚ 1 < a < 2 q™‘‚ „ ‘y‹„ ‰‚€z…Œ’• ‘„• (an ) ‚|Š~† ~Šy‰‚•~ •‘„Š
3
n
‘y‹„ ‰‚€z…Œ’• ‘„• (n3 ) ‡~† •‘„Š ‘y‹„ ‰‚€z…Œ’• ‘„• 2n •†™‘† ann → 0 ‡~† a2n = ( a2 )n → 0 h~†
z”Œ’‰‚
an n3
n3
n
2
2n = an → 0.
P+∞ an P+∞ a n
q›Ž~ ‚•‚†•{ „ €‚–‰‚‘Ž†‡{ •‚†Žy n=1 2n = n=1 ( 2 ) •’€‡ˆ|Š‚† •’Š‚•y€‚‘~† ™‘† ‡~† „ •‚†Žy
P+∞ n3
n=1 2n •’€‡ˆ|Š‚†
P
n~Žy•‚†€‰~
_Š †•”š‚† xn ≥ 0 €†~ ‡y…‚ n •’Š‚•y€‚‘~† ™‘† †•”š‚† x1 +· · ·+xm ≤ +∞
n=1 xn
€†~ ‡y…‚ m
P
nŽ›‘„ ~•™•‚†‹„ X•‘– +∞
n=1 xn = s ∈ R _Š sn = x1 + · · · + xn ‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~
‘„• •‚†Žy• ‘™‘‚ sn → s c•‚†•{ „ ~‡ŒˆŒ’…|~ (sn ) ‚|Š~† ~š‹Œ’•~ †•”š‚† sm ≤ s €†~ ‡y…‚ m
P
b‚š‘‚Ž„ ~•™•‚†‹„ f‚–ŽŒš‰‚ ‘„ •‚†Žy +∞
n=1 an ™•Œ’ ŒŽ|ƒŒ’‰‚
(
xn , ~Š 1 ≤ n ≤ m
an =
0,
~Š n ≥ m + 1
‚•‚†•{ 1 > 0 ‡~† ‚•‚†•{
P+∞
1
n=1 n
q™‘‚ †•”š‚† 0 ≤ an ≤ xn €†~ ‡y…‚ n Œ•™‘‚
P+∞
n=1 an
[‰–•
P+∞
n=1 an
P+∞
= a1 + · · · + am +
n=m+1 an
≤
P+∞
n=1 xn .
= x1 + · · · + xm +
_‹|ƒ‚† Š~ ~•Œ•‚|‹Œ’‰‚ ‘Œ ‚‹{•
nomq_pe
P+∞
n=m+1 0
= x1 + · · · + xm .
m e ‚|Š~† yŽŽ„‘Œ•
_•™•‚†‹„ f~ ”Ž„•†‰Œ•Œ†{•Œ’‰‚ ‘„Š †•™‘„‘~
e=1+
P+∞
1
n=1 n!
~•™ ‘Œ •~Žy•‚†€‰~
X•‘– €†~ Š~ ‡~‘~ˆ{‹Œ’‰‚ •‚ y‘Œ•Œ ™‘† e ∈ Q ‡~† •’€‡‚‡Ž†‰zŠ~ z•‘– e =
_•™ ‘„Š
z”Œ’‰‚
P+∞
1
k!
(k − 1)! l = k! e = k! + k!
n=k+1 n! .
1! + · · · + k! + k!
l
k
™•Œ’ l, k ∈ N
k!
1! ,
k!
. . . , k!
‚|Š~† ~‡zŽ~†Œ• Œ•™‘‚ ‡~† Œ ~Ž†…‰™•
P
P+∞
k!
1
= +∞
A = k! n=k+1 n!
n=k+1 n!
h~…zŠ~• ~•™ ‘Œ’• (k − 1)! l k!
•Žz•‚† Š~ ‚|Š~† ~‡zŽ~†Œ• [‰–•
P
P+∞
1
0 < A = +∞
n=k+1 (k+1)···n <
n=k+1
‡~† ‡~‘~ˆ{€Œ’‰‚ •‚ y‘Œ•Œ
1
(k+1)n−k
=
P+∞
1
n=1 (k+1)n
=
1
1
k+1 1−(1/(k+1))
=
1
k
≤1
p‚†Žz• ‰‚ “…|ŠŒŠ‘‚• ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’•
p‘†• ‚•™‰‚Š‚• •šŒ •ŽŒ‘y•‚†• …~ •Œš‰‚ •šŒ ‡Ž†‘{Ž†~ •š€‡ˆ†•„• €†~ •‚†Žz• ‰‚ “…|ŠŒŠ‘‚• ‰„
~ŽŠ„‘†‡Œš• ™ŽŒ’•
hogqeogm mimhieovj_qmp X•‘– “…|ŠŒ’•~ ~‡ŒˆŒ’…|~ (xn ) ›•‘‚ Š~ †•”š‚† xn ≥ 0 €†~
‡y…‚ n X•‘– ™‘† ’•yŽ”‚† f : [1, +∞)
R t → R “…|ŠŒ’•~ •‘Œ [1, +∞) ›•‘‚ Š~ †•”š‚† f (n) = xn €†~
‡y…‚ n q™‘‚ ’•yŽ”‚† ‘Œ OLPt→+∞ 1 f (u) du „ ‘†‰{ ‘Œ’ ‚|Š~† ~Ž†…‰™• ≥ 0 { +∞ ‡~†
Rt
P
(i) +∞
n=1 xn < +∞ ~Š ‡~† ‰™ŠŒ ~Š OLPt→+∞ 1 f (u) du < +∞
Rt
P
(ii) +∞
n=1 xn = +∞ ~Š ‡~† ‰™ŠŒ ~Š OLPt→+∞ 1 f (u) du = +∞
c•|•„• †•”š‚†
R n+1
Rn
f (u) du ≤ x1 + · · · + xn ≤ x1 + 1 f (u) du
€†~ ‡y…‚ n
1
‡~†
OLPt→+∞
b‚|‘‚ ‘Œ •”{‰~
Rt
1
f (u) du ≤
P+∞
n=1 xn
≤ x1 + OLPt→+∞
Rt
1
f (u) du.
P
_•™•‚†‹„ e •‚†Žy +∞
n=1 xn z”‚† ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• Œ•™‘‚ z”‚† y…ŽŒ†•‰~ ‘Œ Œ•Œ|Œ ‚|Š~† ‰„
~ŽŠ„‘†‡™• ~Ž†…‰™• { +∞ X•‘–
P+∞
n=1 xn = s,
Œ•™‘‚
0 ≤ s ≤ +∞.
_Š sn = x1 + · · · + xn ‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚†Žy• ‘™‘‚
sn → s.
c•‚†•{ „ f ‚|Š~† ‰ŒŠ™‘ŒŠ„ ‚|Š~† ŒˆŒ‡ˆ„Ž›•†‰„ •‚ ‡y…‚ ‡ˆ‚†•‘™ ‡~† “Ž~€‰zŠŒ ’•Œ•†y•‘„‰~ ‘Œ’
[1, +∞) Œ•™‘‚ ŒŽ|ƒ‚‘~† ‘Œ ~™Ž†•‘Œ ŒˆŒ‡ˆ{Ž–‰~
Rt
F (t) = 1 f (u) du
€†~ t ∈ [1, +∞).
p”‚‘†‡y ‰‚ ‘Œ ‡Ž†‘{Ž†Œ ŒˆŒ‡ˆ„Ž›‰~‘Œ• Œ† ‘‚”Š†‡z• z”Œ’Š {•„ ~Š~•‘’”…‚| •‘†• ~•‡{•‚†•
‡~†
X•‘– t ≥ 1 ‡~† n ≥ t c•‚†•{ „ f ‚|Š~† “…|ŠŒ’•~ ‚|Š~† f (t) ≥ f (n) = xn ≥ 0 VŽ~ †•”š‚†
f (t) ≥ 0
VŽ~ ~Š 1 ≤ t′ < t′′ ‘™‘‚ ˆ™€– ‘„•
F (t′′ ) =
R t′′
1
f (u) du =
€†~ ‡y…‚ t ≥ 1.
R t′′
‚|Š~†
R t′
1
t′
f (u) du +
f (u) du ≥ 0 ‡~† ‚•Œ‰zŠ–•
R t′′
t′
f (u) du ≥
R t′
1
f (u) du = F (t′ ).
VŽ~ „ F ‚|Š~† ~š‹Œ’•~ •’ŠyŽ‘„•„ •‘Œ [1, +∞) Œ•™‘‚ ’•yŽ”‚† ‘Œ ™Ž†Œ
Rt
l = OLPt→+∞ F (t) = OLPt→+∞ 1 f (u) du
‡~† „ ‘†‰{ ‘Œ’ ‚|Š~† ~Ž†…‰™• { +∞ nyˆ† ˆ™€– ‘„•
†•”š‚† F (t) =
t ≥ 1 Œ•™‘‚ ‚|Š~† l ≥ 0 b„ˆ~•{
0 ≤ l ≤ +∞.
Rt
1
f (u) du ≥ 0 €†~ ‡y…‚
a†~ ‡y…‚ k ∈ N †•”š‚†
f (k + 1) ≤ f (u) ≤ f (k)
Œ•™‘‚
f (k + 1) =
R k+1
k
f (k + 1) du ≤
R k+1
k
€†~ k ≤ u ≤ k + 1,
f (u) du ≤
‡~† ‚•‚†•{ f (k) = xk ‡~† f (k + 1) = xk+1 ‚|Š~†
xk+1 ≤
nŽŒ•…z‘Œ’‰‚ ‘†• ~Ž†•‘‚Žz• ~Š†•™‘„‘‚•
€†~ k = 1, . . . , n ‡~† •Ž|•‡Œ’‰‚
Rn
x2 + · · · + xn ≤ 1 f (u) du
R k+1
k
R k+1
k
f (k) du = f (k)
f (u) du ≤ xk .
€†~ k = 1, . . . , n − 1 ‡~† ‘†• •‚‹†z• ~Š†•™‘„‘‚•
‡~†
R n+1
1
f (u) du ≤ x1 + · · · + xn ,
~Š‘†•‘Œ|”–• c•‚†•{ sn = x1 + x2 + · · · + xn •’Š‚•y€‚‘~†
R n+1
Rn
f (u) du ≤ sn ≤ x1 + 1 f (u) du.
1
VŽ~ ~•™ ‘~ ™Ž†~
‡~†
•’Š‚•y€‚‘~†
l ≤ s ≤ x1 + l.
q~ (i) (ii) ‚|Š~† y‰‚•‚• •’Šz•‚†‚• ‘„• ‘‚ˆ‚’‘~|~• ~Š†•™‘„‘~•
c|Š~† €Š–•‘{ ~•™ ~•ˆy ‰~…{‰~‘~ ~•‚†ŽŒ•‘†‡Œš ˆŒ€†•‰Œš „ zŠŠŒ†~ ‘Œ’ €‚Š†‡‚’‰zŠŒ’ ŒˆŒ‡ˆ„
Ž›‰~‘Œ•
R +∞
Rt
f (u) du = OLPt→+∞ 1 f (u) du.
1
q~ €‚Š†‡‚’‰zŠ~ ŒˆŒ‡ˆ„Ž›‰~‘~ …~ ‰‚ˆ‚‘„…ŒšŠ •†‚‹Œ•†‡y ~Ž€™‘‚Ž~ •‘Œ ‡‚“yˆ~†Œ
nyŠ‘–•
‰•ŒŽŒš‰‚ ‚•› Š~ €Žy•Œ’‰‚ ‘„Š ‘‚ˆ‚’‘~|~ •†•ˆ{ ~Š†•™‘„‘~ ‘Œ’ ‡Ž†‘„Ž|Œ’ ŒˆŒ‡ˆ„Ž›‰~‘Œ• •‘„
‰ŒŽ“{
R +∞
R +∞
P
f (u) du ≤ +∞
f (u) du
n=1 xn ≤ x1 + 1
1
P
‡~† Š~ •†~‘’•›•Œ’‰‚ ‘Œ ‡‚Š‘Ž†‡™ •’‰•zŽ~•‰~ –• ‚‹{• „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ
R +∞
~Š ‘Œ €‚Š†‡‚’‰zŠŒ ŒˆŒ‡ˆ{Ž–‰~ 1 f (u) du •’€‡ˆ|Š‚†
r•yŽ”‚† ‡~† ‰†~ •~Ž~ˆˆ~€{ ‘Œ’ ‡Ž†‘„Ž|Œ’ ŒˆŒ‡ˆ„Ž›‰~‘Œ• „ Œ•Œ|~ ~Š~“zŽ‚‘~† •‚ •‚†Žz•
‘–Š Œ•Œ|–Š Œ •‚|‡‘„• ~Ž”|ƒ‚† ~•™ ‘ŒŠ ~‡zŽ~†Œ k ~Š‘| ~•™ ‘ŒŠ 1 f~ •‚Ž†ŒŽ†•‘Œš‰‚ •‘Œ Š~ ‘{Š
•†~‘’•›•Œ’‰‚ e ~•™•‚†‹„ ‚|Š~† ‚Š‘‚ˆ›• ™‰Œ†~ ‰‚ ‘„Š ~•™•‚†‹„ •Œ’ {•„ ‡yŠ~‰‚
X•‘– “…|ŠŒ’•~ ~‡ŒˆŒ’…|~ (xn ) ›•‘‚ Š~ †•”š‚† xn ≥ 0 €†~ ‡y…‚ n ≥ k X•‘– ™‘† ’•yŽ”‚† f :
[k, +∞) →
R t R “…|ŠŒ’•~ •‘Œ [k, +∞) ›•‘‚ Š~ †•”š‚† f (n) = xn €†~ ‡y…‚ n ≥ k q™‘‚ ’•yŽ”‚† ‘Œ
OLPt→+∞ k f (u) du „ ‘†‰{ ‘Œ’ ‚|Š~† ~Ž†…‰™• ≥ 0 { +∞ ‡~†
Rt
P
(i) +∞
n=k xn < +∞ ~Š ‡~† ‰™ŠŒ ~Š OLPt→+∞ k f (u) du < +∞
Rt
P
(ii) +∞
n=k xn = +∞ ~Š ‡~† ‰™ŠŒ ~Š OLPt→+∞ k f (u) du = +∞
c•|•„• †•”š‚†
R n+1
Rn
f (u) du ≤ xk + · · · + xn ≤ xk + k f (u) du
€†~ ‡y…‚ n ≥ k
k
‡~†
OLPt→+∞
Rt
k
f (u) du ≤
P+∞
n=k
xn ≤ xk + OLPt→+∞
Rt
k
f (u) du.
e ‘‚ˆ‚’‘~|~ •†•ˆ{ ~Š†•™‘„‘~ €Žy“‚‘~† ‡~† •‘„ ‰ŒŽ“{
R +∞
R +∞
P
f (u) du.
f (u) du ≤ +∞
n=k xn ≤ xk + k
k
P
1
n~Žy•‚†€‰~
f~ ‰‚ˆ‚‘{•Œ’‰‚ ‘†• •Œˆš •„‰~Š‘†‡z• •‚†Žz• +∞
n=1 np m† •‚†Žz• ~’‘z• ‚|Š~†
•„‰~Š‘†‡z• ‡~† •†™‘† ”Ž„•†‰‚šŒ’Š –• •Ž™‘’•~ •š€‡Ž†•„• €†~ •Œˆˆz• yˆˆ‚• •‚†Žz•
P
1
e •‚†Žy +∞
n=1 np z”‚† ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• Œ•™‘‚ z”‚† y…ŽŒ†•‰~ ‘Œ Œ•Œ|Œ ‚|Š~† ‰„ ~ŽŠ„‘†‡™•
~Ž†…‰™• { +∞
P
P+∞
1
_Š p ≤ 0 ‘™‘‚ †•”š‚† n1p ≥ 1 €†~ ‡y…‚ n Œ•™‘‚ +∞
n=1 np ≥
n=1 1 = +∞
1
X•‘– p > 0 q™‘‚ „ ~‡ŒˆŒ’…|~ ( np ) ‚|Š~† “…|ŠŒ’•~ ‡~† z”‚† …‚‘†‡Œš• ™ŽŒ’•
f‚–ŽŒš‰‚ ‘„ •’ŠyŽ‘„•„ u1p „ Œ•Œ|~ ‚|Š~† “…|ŠŒ’•~ •‘Œ [1, +∞) ‡~† •ŽŒ“~Š›• Œ† ‘†‰z• ‘„•
P
1
•‘Œ’• “’•†‡Œš• ‘~’‘|ƒŒŠ‘~† ‰‚ ‘Œ’• ~Š‘|•‘Œ†”Œ’• ™ŽŒ’• ‘„• •‚†Žy• +∞
n=1 np
q›Ž~ ‚|Š~†
Rt 1
Rt 1
t1−p −1
du
=
~Š
p
=
6
1
‡~†
p
1−p
1 u du = ORJ t.
1 u
c•Œ‰zŠ–•
OLPt→+∞
Rt
1
1 up
(
1/(p − 1) < +∞,
du =
+∞,
~Š p > 1
~Š 0 1
n=1 np
= +∞, ~Š p ≤ 1
P
1
c†•†‡›‘‚Ž~ ™•–• z”Œ’‰‚ {•„ •‚† „ ~Ž‰ŒŠ†‡{ •‚†Žy +∞
n=1 n ~•Œ‡ˆ|Š‚† •‘Œ +∞ ‚Š› „ •‚†Žy
P+∞ 1
n=1 n2 •’€‡ˆ|Š‚†
c•†•ˆzŒŠ z”Œ’‰‚ ‡~† ‘†• ”Ž{•†‰‚• ‚‡‘†‰{•‚†•
P+∞ 1
1
1
~Š p > 1,
n=1 np ≤ 1 + p−1
p−1 ≤
ORJ(n + 1) ≤ 1 +
(n+1)1−p −1
1
2p
+ ··· +
1
np
+ ··· +
1
n
√
≤1+
≤ 1 + ORJ n
€†~ ‡y…‚ n,
n1−p −1
1−p
€†~ ‡y…‚ n ~Š 0 ≤ p < 1.
P
1
n~Ž~‘„Ž{•‘‚ •‚ •”z•„ ‰‚ ‘„Š •Ž™‘~•„
™‘† €†~ ‡y…‚ p ‰‚ 0 < p ≤ 1 „ •‚†Žy +∞
n=1 np ‚|Š~†
P+∞
•~Žy•‚†€‰~ •‚†Žy• n=1 xn „ Œ•Œ|~ •‚Š •’€‡ˆ|Š‚† ~ˆˆy €†~ ‘„Š Œ•Œ|~ †•”š‚† xn → 0
√
P
n+1
n~Žy•‚†€‰~
a†~ Š~ ‰‚ˆ‚‘{•Œ’‰‚ ‘„ •š€‡ˆ†•„ ‘„• •‚†Žy• +∞
n=1 2n2 +3 €Žy“Œ’‰‚
1−p
≤1+
1
2
n+1
2n2 +3
Œ•™‘‚
e •‚†Žy
P+∞
1
n=1 n3/2
=
1 1+n−1/2
,
n3/2 2+3n−2
√
n+1 1
3/2
2n2 +3
n
•’€‡ˆ|Š‚† Œ•™‘‚ ‡~† „ •‚†Žy
→ 21 .
P+∞
√
n+1
n=1 2n2 +3
•’€‡ˆ|Š‚†
e ‰‚ˆz‘„ ~’‘›Š ‘–Š •‚†Ž›Š ‰z•– ŒˆŒ‡ˆ„Ž–‰y‘–Š z”‚† {•„ €|Š‚† •‘†• ~•‡{•‚†•
‡~†
n~Žy•‚†€‰~
P
e •‚†Žy +∞
n=1
f‚–ŽŒš‰‚ ‘„ •‚†Žy
√1
n
P+∞
n=1 ORJ(1
+
√1 )
n
~•Œ‡ˆ|Š‚† •‘Œ +∞ Œ•™‘‚ ˆ™€– ‘Œ’ ŒŽ|Œ’
√
ORJ(1+(1/ n))
√
1/ n
‘Œ Œ•Œ|Œ •ŽŒ‡š•‘‚† ~•™ ‘Œ OLPx→0
~•Œ‡ˆ|Š‚† •‘Œ +∞
ORJ(1+x)
x
→ 1,
= 1 •’Š‚•y€‚‘~† ™‘† ‡~† „ •‚†Žy
P+∞
n=1 ORJ(1
+
√1 )
n
hogqeogm prjnrhkvpep qmr &$8&+< X•‘– “…|ŠŒ’•~ ~‡ŒˆŒ’…|~ (xn ) ›•‘‚ Š~ †•”š‚†
xn ≥ 0 €†~ ‡y…‚ n q™‘‚
P+∞ k
P
2 x2k < +∞
(i) +∞
n=1 xn < +∞ ~Š ‡~† ‰™ŠŒ ~Š
Pk=0
P
+∞ k
x
=
+∞
~Š
‡~†
‰™ŠŒ
~Š
(ii) +∞
k=0 2 x2k = +∞
n=1 n
P
P+∞ k
_•™•‚†‹„ m† +∞
n=1 xn ‡~†
k=0 2 x2k z”Œ’Š y…ŽŒ†•‰~ ~“Œš ‚|Š~† •‚†Žz• ‰‚ ‰„ ~ŽŠ„‘†‡Œš•
™ŽŒ’• X•‘–
P+∞
P+∞ k
n=1 xn = s,
k=0 2 x2k = t.
f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn ‡~† tk = x1 + 2x2 + 4x4 + · · · + 2k−1 x2k−1
‘–Š •šŒ •‚†Ž›Š q™‘‚
sn → s,
tk → t
‡~† ‚•|•„• †•”š‚†
sn ≤ s,
tk ≤ t
€†~ ‡y…‚ n ‡~† ‡y…‚ k
X•‘– n ∈ N q™‘‚ ’•yŽ”‚† k0 ∈ Z k0 ≥ 0 ›•‘‚ 2k0 ≤ n < 2k0 +1 c•‚†•{ „ (xn ) ‚|Š~† “…|ŠŒ’•~
‡~† ˆ™€– ‘„• •‚š‘‚Ž„• ~Š†•™‘„‘~•
‚|Š~†
s n = x1 + · · · + xn
= x1 + (x2 + x3 ) + (x4 + x5 + x6 + x7 ) + · · ·
+ (x2k0 −1 + · · · + x2k0 −1 ) + (x2k0 + · · · + xn )
≤ x1 + 2x2 + 4x4 + · · · + 2k0 −1 x2k0 −1 + 2k0 x2k0 = tk0 +1 ≤ t.
VŽ~ €†~ ‡y…‚ n †•”š‚† sn ≤ t Œ•™‘‚ ˆ™€– ‘Œ’ •Ž›‘Œ’ ŒŽ|Œ’
‚|Š~† s ≤ t
c•|•„• •yˆ† ‚•‚†•{ „ (xn ) ‚|Š~† “…|ŠŒ’•~ ‡~† ˆ™€– ‘„• •Ž›‘„• ~Š†•™‘„‘~•
‚|Š~†
tk = x1 + 2x2 + 4x4 + · · · + 2k−1 x2k−1
≤ 2x1 + 2x2 + 2(x3 + x4 ) + · · · + 2(x2k−2 +1 + · · · + x2k−1 )
= 2(x1 + x2 + · · · + x2k−1 ) = 2s2k−1 ≤ 2s.
VŽ~ €†~ ‡y…‚ k ∈ N †•”š‚† tk ≤ 2s Œ•™‘‚ ˆ™€– ‘Œ’ •‚š‘‚ŽŒ’ ŒŽ|Œ’
‚|Š~† t ≤ 2s
VŽ~ s ≤ t ≤ 2s Œ•™‘‚ ‘~ s ‡~† t ‚|Š~† ‚|‘‚ ‡~† ‘~ •šŒ ~Ž†…‰Œ| ‚|‘‚ ‡~† ‘~ •šŒ +∞
P
1
n~Žy•‚†€‰~
f~ ‹~Š~•Œš‰‚ ‘†• •‚†Žz• +∞
n=1 np
P+∞ 1
P+∞
_Š p ≤ 0 ‘™‘‚ n=1 np ≥ n=1 1 = +∞
q›Ž~ z•‘– p > 0 Œ•™‘‚ „ ~‡ŒˆŒ’…|~ ( n1p ) ‚|Š~† “…|ŠŒ’•~ ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’•
c‹‚‘yƒŒ’‰‚ ‘„ •‚†Žy
P+∞ 1 k
P+∞ k 1
k=0 ( 2p−1 ) .
k=0 2 (2k )p =
1
1
Œ•™‘‚ •’€‡ˆ|Š‚† ~Š 2p−1
< 1 ‡~† ~•Œ‡ˆ|Š‚† •‘Œ
e •‚†Žy ~’‘{ ‚|Š~† €‚–‰‚‘Ž†‡{ ‰‚ ˆ™€Œ 2p−1
1
+∞ ~Š 2p−1 ≥ 1
P
1
VŽ~ „ •‚†Žy +∞
n=1 np •’€‡ˆ|Š‚† ~Š p > 1 ‡~† ~•Œ‡ˆ|Š‚† •‘Œ +∞ ~Š p ≤ 1
p ~•†‡y ~Š~•‘š€‰~‘~
X•‘– p ∈ N p ≥ 2 p‘Œ •~Žy•‚†€‰~
~•Œ•‚|‹~‰‚ ™‘† •‚ ‡y…‚ x ∈ [0, 1) ~Š‘†•‘Œ†”‚| „
~‡ŒˆŒ’…|~ (xn ) ‘–Š p ~•†‡›Š •„“|–Š ‘Œ’ x ‰‚ ‘†• ‚‹{• †•†™‘„‘‚• hy…‚ xn ‚|Š~† zŠ~• ~•™ ‘Œ’•
~‡‚Ž~|Œ’• 0, 1, . . . p−1 ‡~† „ ~‡ŒˆŒ’…|~ (xn ) •‚Š ‚|Š~† ‘‚ˆ†‡y •‘~…‚Ž{ p−1 ‡~† ~Š •”„‰~‘|•Œ’‰‚
‘~ ~…ŽŒ|•‰~‘~
sn =
x1
p
+
x2
p2
+ ··· +
xn
pn ,
tn =
x1
p
+
x2
p2
+ ··· +
xn
pn
+
1
pn
€†~ ‡y…‚ n ‘™‘‚ †•”š‚†
sn ≤ x < t n
sn → x,
€†~ ‡y…‚ n,
tn → x.
q›Ž~ ‡yŠŒ’‰‚ ‘„Š ~•ˆ{ •~Ž~‘{Ž„•„ ™‘† ‘~ ~…ŽŒ|•‰~‘~ sn ‚|Š~† ~‡Ž†•›• ‘~ ‰‚Ž†‡y ~…ŽŒ|
•‰~‘~ ‘„• •‚†Žy•
P+∞ xn
n=1 pn .
c•Œ‰zŠ–• ‰•ŒŽŒš‰‚ Š~ •Œš‰‚ ™‘† „ •”z•„ sn → x €Žy“‚‘~† †•Œ•šŠ~‰~
P+∞ xn
n=1 pn = x.
mogpjmp
a‚Š†‡y ”–Ž|• ~Š~“ŒŽy •‚ ‡y•Œ†ŒŠ x ‰†~ ~‡ŒˆŒ’…|~ (xn ) ”~Ž~‡‘„Ž|ƒ‚‘~† ~‡Œ
ˆŒ’…|~ p ~•†‡›Š •„“|–Š ~Š ‡y…‚ xn ‚|Š~† zŠ~• ~•™ ‘Œ’• ~‡‚Ž~|Œ’• 0, 1, . . . p−1 ‡~† „ ~‡ŒˆŒ’…|~
(xn ) •‚Š ‚|Š~† ‘‚ˆ†‡y •‘~…‚Ž{ p − 1
nomq_pe
X•‘– p ∈ N p ≥ 2
P
xn
>~@ X•‘– ~‡ŒˆŒ’…|~ p ~•†‡›Š •„“|–Š (xn ) q™‘‚ „ •‚†Žy +∞
n=1 pn •’€‡ˆ|Š‚† ‡~† ‘Œ y…ŽŒ†•‰y ‘„•
‚|Š~† ~Ž†…‰™• •‘Œ •†y•‘„‰~ [0, 1)
P
xn
>•@ a†~ ‡y…‚ x ∈ [0, 1) ’•yŽ”‚† ‰ŒŠ~•†‡{ ~‡ŒˆŒ’…|~ p ~•†‡›Š •„“|–Š (xn ) ›•‘‚ +∞
n=1 pn = x
_•™•‚†‹„ >~@ e •‚†Žy
P+∞
xn
n=1 pn
z”‚† ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• Œ•™‘‚ z”‚† y…ŽŒ†•‰~ X•‘–
x=
P+∞
xn
n=1 pn .
c•‚†•{ †•”š‚† 0 ≤ xn ≤ p − 1 €†~ ‡y…‚ n ‡~† ‚•†•ˆzŒŠ ‚•‚†•{ †•”š‚† „ €Š{•†~ ~Š†•™‘„‘~
xn ~@ •’Š‚•y€‚‘~†
P+∞ p−1
P
xn
0 ≤ x = +∞
n=1 pn = 1.
n=1 pn <
a†~ ‘„Š ‘‚ˆ‚’‘~|~ †•™‘„‘~ ”Ž„•†‰Œ•Œ†Œš‰‚ ‘ŒŠ ‘š•Œ €†~ ‘Œ y…ŽŒ†•‰~ €‚–‰‚‘Ž†‡{• •‚†Žy• –• ‚‹{•
P+∞ p−1
p−1 P+∞ 1 n−1
1
= p−1
n=1 pn = p
n=1 ( p )
p 1−(1/p) = 1.
>•@ X•‘– 0 ≤ x < 1 X”Œ’‰‚
~•Œ•‚|‹‚† •‘Œ •~Žy•‚†€‰~
™‘† ’•yŽ”‚† ~‡ŒˆŒ’…|~ p
P+∞{•„
xn
~•†‡›Š •„“|–Š (xn ) ›•‘‚ n=1 pn = x f~ ~•Œ•‚|‹Œ’‰‚ ™‘† ~’‘{ ‚|Š~† ‰ŒŠ~•†‡{
_Ž”†‡y …’‰™‰~•‘‚ ™‘† „ ~‡ŒˆŒ’…|~ (xn ) z”‚† •Ž›‘Œ ™ŽŒ x1 = [px] ‡~† Œ† ‚•™‰‚ŠŒ† ™ŽŒ† ‘„•
†‡~ŠŒ•Œ†ŒšŠ ‘ŒŠ ~Š~•ŽŒ‰†‡™ ‘š•Œ
xn+1 = [pn+1 x − pn x1 − · · · − pxn ]
€†~ ‡y…‚ n.
q›Ž~ z•‘– Œ•Œ†~•{•Œ‘‚ ~‡ŒˆŒ’…|~ p ~•†‡›Š •„“|–Š (yn ) ›•‘‚
P+∞ yn
n=1 pn = x.
_•™ ‘„Š
•’Š‚•y€‚‘~†
px = y1 +
P+∞
yn
n=2 pn−1 .
c•‚†•{ †•”š‚† 0 ≤ yn ≤ p − 1 €†~ ‡y…‚ n ≥ 2 ‡~† ‚•†•ˆzŒŠ ‚•‚†•{ †•”š‚† „ €Š{•†~ ~Š†•™‘„‘~
>~@ •’Š‚•y€‚‘~†
yn < p − 1 €†~ ‘Œ’ˆy”†•‘ŒŠ zŠ~Š n ≥ 2 ~•™ ‘„Š •Ž™‘~•„
P+∞ p−1
P+∞ yn
0 ≤ n=2 pn−1 < n=2 pn−1 = 1.
_•™ ‘†•
‡~†
•’Š‚•y€‚‘~†
y1 ≤ px < y1 + 1
‡~† ‚•‚†•{ Œ y1 ‚|Š~† ~‡zŽ~†Œ• z”Œ’‰‚ y1 = [px]
h~‘™•†Š z•‘– m ∈ N _•™ ‘„Š
•’Š‚•y€‚‘~†
P+∞
P
yn
x = m+1
n=m+2
n=1 pn +
yn
pn
‡~† •Œˆˆ~•ˆ~•†yƒŒŠ‘~• ‰‚ ‘ŒŠ pm+1
pm+1 x = pm y1 + · · · + pym + ym+1 +
P+∞
pm+1 x − pm y1 − · · · − pym = ym+1 +
P+∞
{ †•Œ•šŠ~‰~
yn
n=m+2 pn−m−1
yn
n=m+2 pn−m−1 .
h~† •yˆ† ‚•‚†•{ †•”š‚† 0 ≤ yn ≤ p − 1 €†~ ‡y…‚ n ≥ m + 2 ‡~† ‚•‚†•{ †•”š‚† „ €Š{•†~ ~Š†•™‘„‘~
yn ~@ •’Š‚•y€‚‘~†
P
P+∞
p−1
yn
< +∞
0 ≤ n=m+2 pn−m−1
n=m+2 pn−m−1 = 1.
_•™ ‘†•
‡~†
•’Š‚•y€‚‘~†
ym+1 ≤ pm+1 x − pm y1 − · · · − pym < ym+1 + 1
‡~† ‚•‚†•{ Œ ym+1 ‚|Š~† ~‡zŽ~†Œ• z”Œ’‰‚
ym+1 = [pm+1 x − pm y1 − · · · − pym ].
_ˆˆyƒŒŠ‘~• ~•ˆ›• ‘Œ •š‰•ŒˆŒ ‘Œ’ •‚|‡‘„ ~•™ m •‚ n •†~‘’•›ŠŒ’‰‚
‘Œ ~•Œ‘zˆ‚•‰y ‰~• –•
P+∞
‚‹{• Œ•Œ†~•{•Œ‘‚ ~‡ŒˆŒ’…|~ p ~•†‡›Š •„“|–Š (yn ) ‰‚ ‘„Š †•†™‘„‘~ n=1 pynn = x z”‚† •Ž›‘Œ
™ŽŒ y1 = [px] ‡~† Œ† ‚•™‰‚ŠŒ† ™ŽŒ† ‘„• †‡~ŠŒ•Œ†ŒšŠ ‘ŒŠ ~Š~•ŽŒ‰†‡™ ‘š•Œ
yn+1 = [pn+1 x − pn y1 − · · · − pyn ]
€†~ ‡y…‚ n.
n~Ž~‘„Ž{•‘‚ ™‘† ~’‘™• Œ ~Š~•ŽŒ‰†‡™• ‘š•Œ• ‚|Š~† Œ |•†Œ• ‰‚ ‘ŒŠ ~Š~•ŽŒ‰†‡™ ‘š•Œ •Œ’ †‡~ŠŒ
•Œ†ŒšŠ Œ† ™ŽŒ† ‘„• ~Ž”†‡{• (xn )
p’€‡Ž|ŠŒŠ‘~• ‘›Ž~ ‘„Š ~Ž”†‡{ ~‡ŒˆŒ’…|~ (xn ) ‡~† ‘„Š Œ•Œ†~•{•Œ‘‚ yˆˆ„ (yn ) •ˆz•Œ’‰‚ ~‰z
•–• ™‘† ‘~’‘|ƒŒŠ‘~† ‚|Š~† •~“z• ™‘† z”Œ’Š ‘Œ’• |•†Œ’• •Ž›‘Œ’• ™ŽŒ’• y1 = x1 = [px] ‡~† •y•‚†
‘Œ’ ‡Œ†ŠŒš ~Š~•ŽŒ‰†‡Œš ‘š•Œ’ z”Œ’Š ‘Œ’• |•†Œ’• •‚š‘‚ŽŒ’• ™ŽŒ’• ‡~† ‡~‘™•†Š z”Œ’Š ‘Œ’• |•†Œ’•
‘Ž|‘Œ’• ™ŽŒ’• ‡~† ‚•~€–€†‡y z”Œ’Š ‘Œ’• |•†Œ’• n Œ•‘Œš• ™ŽŒ’• €†~ ‡y…‚ n
VŽ~ „ ~‡ŒˆŒ’…|~ p ~•†‡›Š •„“|–Š (xn ) •Œ’P
€Š–Ž|ƒŒ’‰‚ ~•™ ‘Œ •~Žy•‚†€‰~
‚|Š~† „ ‰ŒŠ~
+∞ xn
•†‡{ ~‡ŒˆŒ’…|~ p ~•†‡›Š •„“|–Š (xn ) ›•‘‚ n=1 pn = x
mogpjmp
_Š „ (xn ) ‚|Š~† „ ~‡ŒˆŒ’…|~ ‘–Š p ~•†‡›Š •„“|–Š ‘Œ’ x ∈ [0, 1) „ •‚†Žy
P+∞ xn
ŒŠŒ‰yƒ‚‘~†
p ~•†‡™ ~Šy•‘’€‰~ ‘Œ’ x ‡~† •’Š{…–• ~Š‘†‡~…†•‘Œš‰‚ ~’‘{Š ‘„ •‚†Žy ‰‚
n=1 pn
‘Œ •š‰•ŒˆŒ h0.x1 x2 x3 . . . ip Œ•™‘‚ €Žy“Œ’‰‚
x = h0.x1 x2 x3 . . . ip .
p‘„Š •‚Ž|•‘–•„ p = 10 ”Ž„•†‰Œ•Œ†Œš‰‚ •~Ž~•Œ•†~‡y ‘Œ ~•ˆŒš•‘‚ŽŒ •š‰•ŒˆŒ x = 0.x1 x2 x3 . . .
~Š‘| ‘Œ’ x = h0.x1 x2 x3 . . . i10 .
n~Ž~‘„Ž{•‘‚ ™‘† „ •Ž™‘~•„
ˆz‚† ™‘† ’•yŽ”‚† ~‰“†‰ŒŠŒ•{‰~Š‘„ ~Š‘†•‘Œ†”|~ ~Šy‰‚•~ •‘Œ’•
~Ž†…‰Œš• •‘Œ [0, 1) ‡~† •‘†• ~‡ŒˆŒ’…|‚• p ~•†‡›Š •„“|–Š { †•Œ•šŠ~‰~ ~Šy‰‚•~ •‘Œ’• ~Ž†…‰Œš•
•‘Œ [0, 1) ‡~† •‘~ p ~•†‡y ~Š~•‘š€‰~‘~ h0.x1 x2 x3 . . . ip
mogpjmp
qŒ p ~•†‡™ ~Šy•‘’€‰~ h0.x1 x2 x3 . . . ip ”~Ž~‡‘„Ž|ƒ‚‘~† •‚Ž†Œ•†‡™ ~Š ’•yŽ”Œ’Š
m0 , k0 ›•‘‚ Š~ †•”š‚† xn+k0 = xn €†~ ‡y…‚ n ≥ m0 _’‘™ •„‰~|Š‚† ™‘† ~‰z•–• ‰‚‘y ~•™ ‘Œ ‘‰{‰~
xm0 xm0 +1 . . . xm0 +k0 −1 ‘Œ’ p ~•†‡Œš ~Š~•‘š€‰~‘Œ• ~‡ŒˆŒ’…‚| ‘Œ |•†Œ ‘‰{‰~ ‡~† ~‰z•–• ‰‚‘y
~•™ ~’‘™ ~‡ŒˆŒ’…‚| ‘Œ |•†Œ ‘‰{‰~ ‡~† Œš‘– ‡~… ‚‹{• b„ˆ~•{ ‘Œ p ~•†‡™ ~Šy•‘’€‰~ z”‚† ‘„ ‰ŒŽ“{
h0.x1 . . . xm0 −1 xm0 . . . xm0 +k0 −1 xm0 . . . xm0 +k0 −1 xm0 . . . xm0 +k0 −1 . . . ip .
tŽ„•†‰Œ•Œ†Œš‰‚ ‡~† ‘„ •’Š‘Œ‰Œ€Ž~“|~ h0.x1 . . . xm0 −1 xm0 . . . xm0 +k0 −1 ip .
e •Ž™‘~•„
•‘„Š •‚Ž|•‘–•„ p = 10 •„ˆ~•{ €†~ ‘~ •‚‡~•†‡y ~Š~•‘š€‰~‘~ ‚|Š~† €Š–•‘{
~•™ ‘Œ •„‰Œ‘†‡™ •”Œˆ‚|Œ ”–Ž|• ~•™•‚†‹„ “’•†‡y
c•†•„‰~|ŠŒ’‰‚ ™‘† ‚‡‘™• ~•™ ‘Œ’• ~Ž†…‰Œš• •‘Œ [0, 1) ‡~† Œ† “’•†‡Œ| ~Ž†…‰Œ| z”Œ’Š p ~•†‡y
~Š~•‘š€‰~‘~ _’‘™ ‘Œ ƒ{‘„‰~ ‚Š‘y••‚‘~† •‘Œ •ˆ~|•†Œ ‘„• •‘Œ†”‚†›•Œ’• ~Ž†…‰„‘†‡{•
nomq_pe
X•‘– p ∈ N p ≥ 2 ‡~† x ∈ [0, 1) q™‘‚ Œ x ‚|Š~† Ž„‘™• ~Š ‡~† ‰™ŠŒ ~Š ‘Œ
p ~•†‡™ ‘Œ’ ~Šy•‘’€‰~ ‚|Š~† •‚Ž†Œ•†‡™
_•™•‚†‹„ X•‘– x = h0, x1 . . . xm0 −1 xm0 . . . xm0 +k0 −1 ip
q™‘‚
xm0 −1
pm0 −1
xm0 −1
pm0 −1
x=
x1
p
=
x1
p
=
x1 pm0 −2 +···+xm0 −1
pm0 −1
+ ··· +
+ ··· +
+
+
+
xm0
pm 0
xm0
pm 0
+ ··· +
+ ··· +
xm0 +k0 −1 1
1 + pk10 + p2k
0
pm0 +k0 −1
xm0 +k0 −1 1
pm0 +k0 −1 1−(1/pk0 )
+ ···
xm0 pk0 −1 +···+xm0 +k0 −1
,
pm0 −1 (pk0 −1)
Œ•™‘‚ ‚|Š~† “~Š‚Ž™ ™‘† Œ x ‚|Š~† Ž„‘™•
_Š‘†•‘Ž™“–• z•‘– ™‘† Œ x ∈ [0, 1) ‚|Š~† Ž„‘™• •„ˆ~•{ x = ab ™•Œ’ a, b ∈ Z 0 ≤ a < b
aŽy“Œ’‰‚ p = p1 n1 · · · pr nr ™•Œ’ p1 , . . . , pr ‚|Š~† Œ† •Ž›‘Œ† •~Žy€ŒŠ‘‚• ‘Œ’ p ‡~† n1 , . . . , nr ∈
N m‰Œ|–• €Žy“Œ’‰‚ b = p1 l1 · · · pr lr b′ ™•Œ’ l1 , . . . , lr ∈ Z l1 , . . . , lr ≥ 0 ~Š ‡y•Œ†Œ• pj •‚Š
‚|Š~† •Ž›‘Œ• •~Žy€–Š ‘Œ’ b ‘™‘‚ Œ ~Š‘|•‘Œ†”Œ• lj ‚|Š~† 0 ‡~† Œ b′ ∈ N ‚|Š~† •”‚‘†‡y •Ž›‘Œ• ‰‚
‘ŒŠ p
f‚–ŽŒš‰‚ Œ•Œ†ŒŠ•{•Œ‘‚ m0 ∈ N ›•‘‚ Š~ †•”š‚† (m0 − 1)nj ≥ lj €†~ ‡y…‚ j = 1, . . . , r
mŽ|ƒŒ’‰‚ vj = (m0 − 1)nj − lj Œ•™‘‚ vj ∈ Z vj ≥ 0 q™‘‚
x=
a
b
=
a
p1 l1 ···pr lr b′
=
ap1 v1 ···pr vr
(p1 n1 ···pr nr )m0 −1 b′
=
a′
,
pm0 −1 b′
™•Œ’ a′ ∈ Z a′ ≥ 0
q›Ž~ •†~†ŽŒš‰‚ ‘Œ’• p, p2 , p3 , . . . ‰‚ ‘ŒŠ b′ q~ •†…~Šy ’•™ˆŒ†•~ ~’‘›Š ‘–Š •†~†Žz•‚–Š •„ˆ~•{
Œ† 0, . . . , b′ − 1 ‚|Š~† •‚•‚Ž~•‰zŠ~ ~ˆˆy Œ† ~Ž†…‰Œ| ‚|Š~† y•‚†ŽŒ† Œ•™‘‚ ‘Œ’ˆy”†•‘ŒŠ •šŒ ~•™
~’‘Œš• …~ •›•Œ’Š ‘Œ |•†Œ ’•™ˆŒ†•Œ ™‘~Š •†~†Ž‚…ŒšŠ ‰‚ ‘ŒŠ b′ b„ˆ~•{ ’•yŽ”Œ’Š t, s ∈ N t < s
›•‘‚ pt = qt b′ + z ‡~† ps = qs b′ + z ™•Œ’ qt , qs ∈ Z ‡~† z ∈ {0, . . . , b′ − 1} p’Š‚•y€‚‘~†
pt (ps−t − 1) = ps − pt = (qs − qt )b′ Œ•™‘‚ Œ b′ •†~†Ž‚| ‘ŒŠ pt (ps−t − 1) c•‚†•{ Œ† b′ , p ‚|Š~†
•”‚‘†‡y •Ž›‘Œ† Œ b′ •†~†Ž‚| ‘ŒŠ ps−t − 1 Œ•™‘‚ ’•yŽ”‚† b′′ ∈ N ›•‘‚ b′ b′′ = ps−t − 1 mŽ|ƒŒ’‰‚
‘ŒŠ k0 = s − t ∈ N ‡~† z”Œ’‰‚ ™‘† b′ b′′ = pk0 − 1 ‡~† ‚•Œ‰zŠ–•
x=
a′ b′′
pm0 −1 b′ b′′
=
a′′
,
pm0 −1 (pk0 −1)
™•Œ’ a′′ = a′ b′′ ∈ Z ‡~† 0 ≤ a′′ < pm0 −1 (pk0 − 1) qŒ ‘‚ˆ‚’‘~|Œ †•”š‚† •†™‘† 0 ≤ x < 1
h~‘™•†Š ‚‡‘‚ˆŒš‰‚ ‘„ •†~|Ž‚•„ ‘Œ’ a′′ ‰‚ ‘ŒŠ pk0 − 1 Œ•™‘‚
a′′ = w(pk0 − 1) + u,
b‚|‘‚ ‘„Š y•‡„•„
™•Œ’ w ∈ Z 0 ≤ w < pm0 −1 ‡~† u ∈ {0, . . . , pk0 − 2}
qzˆŒ• €Žy“Œ’‰‚ ‘~ p ~•†‡y ~Š~•‘š€‰~‘~ •‚|‘‚ ‘„Š y•‡„•„
w = x1 pm0 −2 + · · · + xm0 −1 ,
‘–Š w u •‘„ ‰ŒŽ“{
u = xm0 pk0 −1 + · · · + xm0 +k0 −1
‡~† •~Ž~‘„ŽŒš‰‚ ™‘† Œ† xm0 , . . . , xm0 +k0 −1 •‚Š ‚|Š~† ™ˆŒ† |•Œ† ‰‚ p − 1 •†™‘† ~ˆˆ†›• …~ {‘~Š
u = (p − 1)pk0 −1 + · · · + (p − 1)p + (p − 1) = pk0 − 1 p’‰•‚Ž~|ŠŒ’‰‚ ™‘†
x=
=
x1 pm0 −2 +···+xm0 −1
w(pk0 −1)+u
u
= pmw0 −1 + pm0 −1 (p
k0 −1) =
pm0 −1
pm0 −1 (pk0 −1)
xm0 +k0 −1 1
xm0 −1
xm0
x1
p + · · · + pm0 −1 + pm0 + · · · + pm0 +k0 −1 1− k1
+
xm0 pk0 −1 +···+xm0 +k0 −1
pm0 −1 (pk0 −1)
p 0
=
x1
p
+ ··· +
xm0 −1
pm0 −1
+
xm0
pm 0
+ ··· +
xm0 +k0 −1 pm0 +k0 −1
1+
1
pk0
+
1
p2k0
= hx1 . . . xm0 −1 xm0 . . . xm0 +k0 −1 ip .
+ ···
VŽ~ Œ x z”‚† •‚Ž†Œ•†‡™ p ~•†‡™ ~Šy•‘’€‰~
_•‡{•‚†•
tŽ„•†‰Œ•Œ†›Š‘~• ‘„ •‚†Žy
P+∞
1
n=1 n2
•’€‡ˆ|Š‚†
P+∞
1
n=1 n(n+1)
~•™ ‘„Š y•‡„•„
~•Œ•‚|‹‘‚ ™‘† „ •‚†Žy
P
n2 +3n+1
p’€‡Ž|ŠŒŠ‘~• ‰‚ ~•ˆŒš•‘‚Ž‚• •‚†Žz• ‰‚ˆ‚‘{•‘‚ –• •ŽŒ• ‘„ •š€‡ˆ†•„ ‘†• +∞
n=1 n4 −n2 +4
√
P+∞
P+∞ n n+2n+1 P+∞ 1 P+∞
P+∞ √
1
2
√
√
ORJ(1 + n12 )
n=1
n=1
n=1
n=1 n n n
n=1 ( 1 + n − n)
2n2 +1
n(n+1)(n+2)
P+∞ √n+1−√n P+∞ 1
P+∞
1 P+∞ 1/n
1 P+∞
2n+1
VLQ
)
−
1
(e
−
1)
n(1
−
FRV
n
ORJ
n=1
n=1
n=1
n=1
n=1
n
n
n
2n−1
P+∞ (n+1)nn
n=1 nn+1
P
a √1 − √ 1
‡~†
`Ž‚|‘‚ ‘†• ‘†‰z• ‘Œ’ a €†~ ‘†• Œ•Œ|‚• ‡~…‚‰†y ~•™ ‘†• •‚†Žz• +∞
n=1 n
n
n+1
√
P+∞ a √
√
n=1 n ( n + 1 − 2 n + n − 1) •’€‡ˆ|Š‚†
P
1
`Ž‚|‘‚ ‘†• ‘†‰z• ‘–Š a b ‰‚ a > b > 0 €†~ ‘†• Œ•Œ|‚• ‡~…‚‰†y ~•™ ‘†• •‚†Žz• +∞
n=2 na −nb ‡~†
P+∞ 1
n=1 an −bn •’€‡ˆ|Š‚†
P
an
`Ž‚|‘‚ ‘†• ‘†‰z• ‘–Š a, b, c > 0 €†~ ‘†• Œ•Œ|‚• „ •‚†Žy +∞
n=1 bn +cn •’€‡ˆ|Š‚†
_•Œ•‚|‹‘‚ ™‘† „ •‚†Žy
1·3·5···(2n−1)
n=1 2·5·8···(3n−1)
P+∞
•’€‡ˆ|Š‚†
P+∞ n P+∞
P
1√
1
c“~Ž‰™•‘‚ ‘Œ ŒˆŒ‡ˆ„Ž–‘†‡™ ‡Ž†‘{Ž†Œ •‘†• +∞
n=1 n2 +1
n=1 (n+1)( n+1)
n=1 n2 +1
P+∞
P+∞ −n P+∞ en P+∞ 1
P+∞
P+∞
1
1
√ 1
n=2 n(ORJ n)2
n=1 ne
n=1 1+e2n
n=1
n=2 n ORJ n
n=3 n ORJ n ORJ(ORJ n)
n(n+1)
P+∞
1
n=3 n ORJ n(ORJ(ORJ n))2 a†~ ™•‚• •‚†Žz• •’€‡ˆ|ŠŒ’Š •Ž‚|‘‚ ‚‡‘†‰{•‚†• €†~ ‘Œ y…ŽŒ†•‰y ‘Œ’• a†~
™•‚• •‚†Žz• ~•Œ‡ˆ|ŠŒ’Š •‘Œ +∞ •Ž‚|‘‚ ‚‡‘†‰{•‚†• €†~ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘y ‘Œ’• h~‘™•†Š ‚“~Ž
‰™•‘‚ ‡~† ‘Œ ‡Ž†‘{Ž†Œ •’‰•š‡Š–•„• •‘†• •~Ž~•yŠ– •‚†Žz•
P
P+∞
1
1
_•Œ•‚|‹‘‚ ™‘† Œ† •‚†Žz• +∞
n=2 n(ORJ n)p
n=3 n ORJ n(ORJ(ORJ n))p •’€‡ˆ|ŠŒ’Š ~Š p > 1 ‡~†
~•Œ‡ˆ|ŠŒ’Š •‘Œ +∞ ~Š p ≤ 1
P
1
_•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n=2 na (ORJ n)b •’€‡ˆ|Š‚† ~Š a = 1 b > 1 ‡~† ~Š a > 1 ‡~† ™‘†
~•Œ‡ˆ|Š‚† •‘Œ +∞ •‚ ‡y…‚ yˆˆ„ •‚Ž|•‘–•„
P
1
X•‘– a > 0 _•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n=1 an nb •’€‡ˆ|Š‚† ~Š a = 1 b > 1 ‡~† ~Š a > 1 ‡~† ™‘†
~•Œ‡ˆ|Š‚† •‘Œ +∞ •‚ ‡y…‚ yˆˆ„ •‚Ž|•‘–•„
P
1
X•‘– a > 0 _•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n=2 an (ORJ n)b •’€‡ˆ|Š‚† ~Š a > 1 ‡~† ™‘† ~•Œ‡ˆ|Š‚† •‘Œ
+∞ •‚ ‡y…‚ yˆˆ„ •‚Ž|•‘–•„
b†~‘’•›•‘‚ ‡~† ~•Œ•‚|‹‘‚ ‰†~ •~Ž~ˆˆ~€{ ‘Œ’ ‡Ž†‘„Ž|Œ’ •’‰•š‡Š–•„• …‚–Ž›Š‘~• ‘†• •’
Šy‰‚†• pk ‘Œ’ Œ•Œ†Œ’•{•Œ‘‚ “’•†‡Œš p ≥ 3 ~Š‘| ‘–Š •’Šy‰‚–Š 2k ‘Œ’ 2
X•‘– p > 1
P+∞ 1
1
1
1
_•Œ•‚|‹‘‚ ™‘† †•”š‚† (p−1)k
p−1 ≤
n=k np ≤ kp + (p−1)kp−1 €†~ ‡y…‚ k
P
1
1
_•Œ•‚|‹‘‚ ™‘† k p−1 +∞
n=k np → p−1 ™‘~Š k → +∞
P
1
_•Œ•‚|‹‘‚ ™‘† †•”š‚† OLPp→1+ (p − 1) +∞
n=k np = 1 €†~ ‡y…‚ k
P
x
x
X•‘– x > 0 _•Œ•‚|‹‘‚ ™‘† π2 − DUFWDQ x1 ≤ +∞
n=1 n2 +x2 ≤ 1+x2 +
P+∞ x
_•Œ•‚|‹‘‚ ™‘† OLPx→+∞ n=1 n2 +x2 = π2
π
2
− DUFWDQ x1
P+∞
VŽ~
P+∞‘| ~•~Š‘y‰‚ ~Š ‡y•Œ†Œ• †•”’Ž†•‘‚| ™‘† €‚Š†‡y †•”š‚† „ ‚Š~ˆˆ~€{ OLPx→+∞ n=1 fn (x) =
n=1 OLPx→+∞ fn (x) ‘–Š •’‰•™ˆ–Š ‘Œ’ ŒŽ|Œ’ ‡~† ‘„• •‚†Žy•
P
X•‘– ™‘† P
†•”š‚† xn >P0 €†~ ‡y…‚ n ‡~† a ≥ 1 _Š „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ~•Œ•‚|‹‘‚
a
+∞ xn
a
x
™‘† ‡~† Œ† •‚†Žz• +∞
•’€‡ˆ|ŠŒ’Š
n=1 1+xn a
n=1 n
X•‘– ™‘† †•”š‚† xn ≥ 0 €†~ ‡y…‚ n
P
P+∞ √xn
_Š „ •‚†Žy +∞
x
•’€‡ˆ|Š‚†
~•Œ•‚|‹‘‚
™‘†
‡~†
„
•‚†Žy
n
n=1 n •’€‡ˆ|Š‚†
Pn=1
P
√
+∞
_Š „ •‚†Žy n=1 xn •’€‡ˆ|Š‚† ~•Œ•‚|‹‘‚ ™‘† ‡~† „ •‚†Žy +∞
n=1 xn xn+1 •’€‡ˆ|Š‚† _Š ‚•†
•ˆzŒŠ „ (xn ) ‚|Š~† “…|ŠŒ’•~
P ‘™‘‚ ~•Œ•‚|‹‘‚ ‡~† ‘Œ ~Š‘|•‘ŽŒ“Œ
P+∞ √
`Ž‚|‘‚ •~Žy•‚†€‰~ •‚†Žy• +∞
n=1 xn „ Œ•Œ|~ ~•Œ‡ˆ|Š‚† •‘Œ +∞ ›•‘‚ „ •‚†Žy
n=1 xn xn+1
Š~ •’€‡ˆ|Š‚†
>~@ X•‘– ™‘† †•”š‚† xn , yn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† †•”š‚† ‘‚ˆ†‡y xxn+1
≤ yn+1
yn _Š „
n
P+∞
P+∞
•‚†Žy n=1 yn •’€‡ˆ|Š‚† ~•Œ•‚|‹‘‚ ™‘† ‡~† „ •‚†Žy n=1 xn •’€‡ˆ|Š‚†
>•@ X•‘– ™‘† †•”š‚† xn > 0 €†~ ‡y…‚ n
P
_Š 0 < a < 1 ‡~† ~Š †•”š‚† ‘‚ˆ†‡y xxn+1
xn •’€‡ˆ|Š‚†
≤ a ~•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n
P+∞ n=1
xn+1
_Š a > 1 ‡~† ~Š †•”š‚† ‘‚ˆ†‡y xn ≥ a ~•Œ•‚|‹‘‚ ™‘† „ •‚†Žy n=1 xn ~•Œ‡ˆ|Š‚†
>€@ X•‘– ™‘† †•”š‚† xn > 0 €†~ ‡y…‚ n
P
1 a
_Š a > 1 ‡~† ~Š †•”š‚† ‘‚ˆ†‡y xxn+1
) ~•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
xn •’€‡ˆ|Š‚†
≤ (1 − n+1
n
Pn=1
xn+1
+∞
1 a
_Š a ≤ 1 ‡~† ~Š †•”š‚† ‘‚ˆ†‡y xn ≥ (1 − n+1 ) ~•Œ•‚|‹‘‚ ™‘† „ •‚†Žy n=1 xn ~•Œ‡ˆ|Š‚†
P
>~@ X•‘– (xn ) “…|ŠŒ’•~ ›•‘‚ Š~ †•”š‚† xn ≥ 0 €†~ ‡y…‚ n _Š +∞
n=1 xn < +∞ ~•Œ•‚|‹‘‚
™‘† nxn → 0
P
1
_•Œ•‚|‹‘‚ ™‘† +∞
n=1 np = +∞ ~Š 0 ≤ p ≤ 1
>•@ b‚|‘‚ ‘„Š y•‡„•„
X•‘– ~‡ŒˆŒ’…|~ (xn ) ›•‘‚ Š~ †•”š‚† xn+1 ≤
P
‡~† xn → 0 _•Œ•‚|‹‘‚ ™‘† +∞
n=1 n(xn − 2xn+1 + xn+2 ) = x1
xn +xn+2
2
€†~ ‡y…‚ n
X•‘– ™‘†
€†~ ‡y…‚ n _Š „ (xnk ) ‚|Š~† Œ•Œ†~•{•Œ‘‚ ’•Œ~‡ŒˆŒ’…|~ ‘„• (xn )
P †•”š‚† xn ≥
P0+∞
~•Œ•‚|‹‘‚ ™‘† +∞
≤
x
k=1 nk
n=1 xn
P+∞
P
X•‘– ™‘† †•”š‚† xnP
≤ yn ≤ zn €†~ ‡y…‚ n _Š Œ† •‚†Žz• +∞
n=1 zn •’€‡ˆ|ŠŒ’Š
n=1 xn ‡~†
~•Œ•‚|‹‘‚ ™‘† ‡~† „ •‚†Žy +∞
y
•’€‡ˆ|Š‚†
n=1 n
X•‘– m1 m2 m3 , . . . ‡~‘y €Š„•|–• ~š‹Œ’•~ •†y‘~‹„ Œ† “’•†‡Œ| Œ†P
Œ•Œ|Œ† •‚Š •‚Ž†z”Œ’Š
1
‘Œ •‚‡~•†‡™ •„“|Œ 3 •‘Œ •‚‡~•†‡™ ‘Œ’• ~Šy•‘’€‰~ _•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n=1 mn •’€‡ˆ|Š‚†
P
™‘† †•”š‚† xn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† fz‘Œ’‰‚
PX•‘–
rn = +∞
x
€†~
‡y…‚
n
k=n k
rn+1
xn
1
m+1
_Š p ≥ 1 ~•Œ•‚|‹‘‚ ™‘† †•”š‚† rxm+1
p + ··· + r p ≥ r
p €†~ ‡y…‚ m, n ‰‚ m < n
p−1 − r
n
m+1
m+1
e •’Šz”‚†~ •‘„Š y•‡„•„
‡~† •’‰•‚ŽyŠ~‘‚ ™‘† „ •‚†Žy
P+∞
xn
n=1 rn p ~•Œ‡ˆ|Š‚†
1
†•”š‚† rxnnp ≤ 1−p
(rn 1−p
_Š 0 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† „ •‚†Žy
sn = nk=1 xk €†~ ‡y…‚ n
P
xn
_•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n=1 1+xn ~•Œ‡ˆ|Š‚†
m+1
_Š p ≤ 1 ~•Œ•‚|‹‘‚ ™‘† †•”š‚† sxm+1
p + ··· +
P+∞ xn
•’‰•‚ŽyŠ~‘‚ ™‘† „ •‚†Žy n=1 sn p ~•Œ‡ˆ|Š‚†
_Š p > 1 ~•Œ•‚|‹‘‚ ™‘† †•”š‚†
P
xn
„ •‚†Žy +∞
n=1 sn p •’€‡ˆ|Š‚†
xn
sn p
≤
xn
sn p
1
1
p−1 sn−1 p−1
≥
1
sn p−1
−
P+∞
n=1 xn
sm
sn p
~•Œ‡ˆ|Š‚† fz‘Œ’‰‚
€†~ ‡y…‚ m, n ‰‚ m < n ‡~†
1
€†~ ‡y…‚ n ≥ 2 ‡~† •’‰•‚ŽyŠ~‘‚ ™‘†
− sn p−1
P
>~@ X•‘– ™‘† †•”š‚† xn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘†P
„ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† _•Œ•‚|‹‘‚
™‘† ’•yŽ”‚† ~‡ŒˆŒ’…|~ (yn ) ›•‘‚ yn → +∞ ‡~† „ •‚†Žy +∞
x
y
n=1 n n Š~ •’€‡ˆ|Š‚†
P+∞
>•@ X•‘– ™‘† †•”š‚† xn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘†
„
•‚†Žy
n=1 xn ~•Œ‡ˆ|Š‚† _•Œ•‚|‹‘‚ ™‘†
P+∞
’•yŽ”‚† ~‡ŒˆŒ’…|~ (yn ) ›•‘‚ yn → 0 ‡~† „ •‚†Žy n=1 xn yn Š~ ~•Œ‡ˆ|Š‚†
>€@ X•‘– ™‘† „ (yn ) ‚|Š~† €Š„•|–• ~š‹Œ’•~ ‡~†
n ) ›•‘‚ Š~
Pyn → +∞ _•Œ•‚|‹‘‚ ™‘† ’•yŽ”‚†P(x
+∞
†•”š‚† xn > 0 €†~ ‡y…‚ n ‡~† ›•‘‚ „ •‚†Žy +∞
x
Š~
•’€‡ˆ|Š‚†
‡~†
„
•‚†Žy
n
n=1
n=1 xn yn Š~
~•Œ‡ˆ|Š‚†
>•@ X•‘– ™‘† „ (yn ) ‚|Š~† €Š„•|–• “…|ŠŒ’•~ P
‡~† yn → 0 _•Œ•‚|‹‘‚ ™‘† ’•yŽ”‚†P(xn ) ›•‘‚ Š~
+∞
†•”š‚† xn > 0 €†~ ‡y…‚ n ‡~† ›•‘‚ „ •‚†Žy +∞
n=1 xn Š~ ~•Œ‡ˆ|Š‚† ‡~† „ •‚†Žy
n=1 xn yn Š~
•’€‡ˆ|Š‚†
X•‘– an , bn ≥ 0 €†~ ‡y…‚ n ‡~† p, q > 1 ›•‘‚ p1 + 1q = 1
P
P+∞ p
q
>~@ _Š n=1 an < +∞ ‡~† +∞
n=1 bn < +∞ ~•Œ•‚|‹‘‚ ‘„Š ~Š†•™‘„‘~ ‘Œ’ +¶OGHU €†~ •‚†Žz•
P+∞
n=1 an bn
≤
P+∞
n=1 an
p 1/p
P+∞
n=1 bn
q 1/q .
_•Œ•‚|‹‘‚ ™‘† †•”š‚† „ †•™‘„‘~ ~Š ‡~† ‰™ŠŒ ~Š ’•yŽ”Œ’Š s, t ≥ 0 ™”† ‡~† Œ† •šŒ |•Œ† ‰‚ 0 ›•‘‚ Š~
†•”š‚†
san p = tbn q €†~ ‡y…‚
P+∞
Pn
1
1
2
_Š n=1 an 2 < +∞ ‡~† +∞
n=1 bn < +∞ ‚•‚†•{ 2 + 2 = 1 „ •Œˆš •„‰~Š‘†‡{ ~Š†•™‘„‘~ ‘Œ’
&DXFK\
P+∞
P+∞ 2 1/2 P+∞ 2 1/2
,
n=1 bn
n=1 an bn ≤
n=1 an
‚|Š~† ‚†•†‡{ •‚Ž|•‘–•„ ‘„• ~Š†•™‘„‘~• ‘Œ’ +¶OGHU
P
P+∞ p
p
>•@ _Š +∞
n=1 an < +∞ ‡~†
n=1 bn < +∞ ~•Œ•‚|‹‘‚ ‘„Š ~Š†•™‘„‘~ ‘Œ’ 0LQNRZVNL €†~
•‚†Žz•
P+∞
P+∞ p 1/p
P+∞ p 1/p
p 1/p ≤
+
.
n=1 (an + bn )
n=1 an
n=1 bn
_•Œ•‚|‹‘‚ ™‘† †•”š‚† „ †•™‘„‘~ ~Š ‡~† ‰™ŠŒ ~Š ’•yŽ”Œ’Š s, t ≥ 0 ™”† ‡~† Œ† •šŒ |•Œ† ‰‚ 0 ›•‘‚ Š~
†•”š‚† san = tbn €†~ ‡y…‚ n
P
`Ž‚|‘‚ •’ŠyŽ‘„•„ f : [1, +∞) → [0, +∞) •’Š‚”{ •‘Œ [1, +∞) ›•‘‚ +∞
n=1 f (n) = +∞
Rt
‡~† ‘Œ OLPt→+∞ 1 f (u) du Š~ ‚|Š~† ~Ž†…‰™•
P
`Ž‚|‘‚ •’ŠyŽ‘„•„ f : [1, +∞) → [0, +∞) •’Š‚”{ •‘Œ [1, +∞) ›•‘‚ +∞
n=1 f (n) < +∞ ‡~†
Rt
OLPt→+∞ 1 f (u) du = +∞
m† ~Š†•™‘„‘‚• ‘„• y•‡„•„•
€†~ •‚†Žz• e y•‡„•„ ~’‘{ •’Š‚”|ƒ‚‘~† •‘„Š y•‡„•„
p‘Œ …‚›Ž„‰~
‚|•~‰‚ ™‘† €†~ ‡y…‚ •‚†Žy ‰„ ~ŽŠ„‘†‡›Š ™Ž–Š „ ~‡ŒˆŒ’…|~ ‘–Š ‰‚Ž†‡›Š
~…ŽŒ†•‰y‘–Š ‘„• ‚|Š~† ~š‹Œ’•~ ~‡ŒˆŒ’…|~ ‰„ ~ŽŠ„‘†‡›Š ~Ž†…‰›Š p‚ •’Š•’~•‰™ ‰‚ ‘„Š y•‡„•„
~•Œ•‚|‹‘‚ ™‘† ~Š‘†•‘Ž™“–• •‚ ‡y…‚ ~š‹Œ’•~ ~‡ŒˆŒ’…|~ ‰„ ~ŽŠ„‘†‡›Š ~Ž†…‰›Š ~Š‘†•‘Œ†
”‚| ‰†~ •‚†Žy ‰„ ~ŽŠ„‘†‡›Š ™Ž–Š z‘•† ›•‘‚ „ ~‡ŒˆŒ’…|~ ~’‘{ Š~ ‘~’‘|ƒ‚‘~† ‰‚ ‘„Š ~‡ŒˆŒ’…|~ ‘–Š
‰‚Ž†‡›Š ~…ŽŒ†•‰y‘–Š ‘„• •‚†Žy•
`Ž‚|‘‚ ‘Œ •’~•†‡™ ‘Œ ‘‚‘Ž~•†‡™ ‡~† ‘Œ •‚‡~‚‹~•†‡™ ~Šy•‘’€‰~ ‘–Š
√
z‡‘„ •‚‡~•†‡{ ‡~† ‘„Š z‡‘„ •’~•†‡{ •ŽŒ•z€€†•„ ‘Œ’ 2 − 1
7
16
31
32
`Ž‚|‘‚ ‘„Š
X•‘– p ∈ N p ≥ 2 ‡~† x, y ∈ [0, 1) _Š €†~ ‡y•Œ†ŒŠ n Œ† n Œ•‘z• p ~•†‡z• •ŽŒ•‚€€|•‚†•
‘–Š x, y ‚|Š~† |•†‚• ~•Œ•‚|‹‘‚ ™‘† |x − y| < p1n
X•‘– p ∈ N p ≥ 2 ‡~† x ∈ [0, 1) _Š sn ‚|Š~† „ n Œ•‘{ p ~•†‡{ •ŽŒ•z€€†•„ ‘Œ’ x •Œ†y
‚|Š~† ‘~ p ~•†‡y ~Š~•‘š€‰~‘~ ‘–Š x − sn pn (x − sn )
X•‘– p ∈ N p ≥ 2 ‡~† x, y ∈ [0, 1)
_Š x + y < 1 ~•Œ•‚|‹‘‚ ™‘† ‘Œ •“yˆ‰~ •‘ŒŠ ’•ŒˆŒ€†•‰™ ‘Œ’ x + y ‰‚ ‘„Š ~Š‘†‡~‘y•‘~•„ ‘–Š
x, y ~•™ ‘†• n Œ•‘z• p ~•†‡z• •ŽŒ•‚€€|•‚†• ‘Œ’• ‚|Š~† < p2n
_•Œ•‚|‹‘‚ ™‘† ‘Œ ~Š‘|•‘Œ†”Œ •“yˆ‰~ •‘ŒŠ ’•ŒˆŒ€†•‰™ ‘Œ’ xy ‚|Š~† <
2
pn
−
1
p2n
X•‘– p ∈ N p ≥ 2 ‡~† Œ•Œ†Œ•{•Œ‘‚ •’€‡‚‡Ž†‰zŠŒ p ~•†‡™ •„“|Œ k •„ˆ~•{ 0 ≤ k ≤
p−1 _•Œ•‚|‹‘‚ ™‘† ‘Œ •šŠŒˆŒ ‘–Š ~Ž†…‰›Š •‘Œ [0, 1) ‘–Š Œ•Œ|–Š ‘Œ n Œ•‘™ p ~•†‡™ •„“|Œ ‚|Š~†
|•Œ ‰‚ k ‚|Š~† „ zŠ–•„ pn−1 •†~•‘„‰y‘–Š ‘š•Œ’ [a, b) nŒ†y ~‡Ž†•›• ‚|Š~† ~’‘y ‘~ •†~•‘{‰~‘~
‡~† ‘| ‰{‡Œ• z”‚† ‡~…zŠ~ ~•™ ~’‘y nŒ†™ ‚|Š~† ‘Œ •’ŠŒˆ†‡™ ‰{‡Œ• ~’‘›Š ‘–Š •†~•‘„‰y‘–Š
X•‘– p ∈ N p ≥ 2 ‡~† x ∈ [0, 1) _•Œ•‚|‹‘‚ ™‘† ‘~ p ~•†‡y •„“|~ ‘Œ’ x ‚|Š~† ‘‚ˆ†‡y 0 ~Š
‡~† ‰™ŠŒ ~Š x = m
n ™•Œ’ m ∈ Z n ∈ N 0 ≤ m < n JFG(m, n) = 1 ‡~† Œ† •Ž›‘Œ† •~Žy€ŒŠ‘‚•
‘Œ’ n ‚|Š~† •Ž›‘Œ† •~Žy€ŒŠ‘‚• ‡~† ‘Œ’ p
X•‘– p ∈ N p ≥ 2 _•Œ•‚|‹‘‚ ™‘† €†~ ‡y…‚ x ∈ N ’•yŽ”Œ’Š ‰ŒŠ~•†‡Œ| n0 ∈ Z n0 ≥ 0
‡~† x0 , x1 , . . . , xn0 ∈ Z ‰‚ 0 ≤ x0 , x1 , . . . , xn0 ≤ p − 1 ‡~† xn0 ≥ 1 ›•‘‚ x = xn0 pn0 + · · · +
x1 p + x0
c“~Ž‰™•‘‚ •‘Œ’• ~Ž†…‰Œš• 2 16 354 10385 ‰‚ p = 10, 2, 3, 16
r•ŒˆŒ€|•‘‚ ‘Œ’• ~Ž†…‰Œš• h0.34239239239239 . . . i10 h0.101101101101101 . . . i2 ‡~†
h0.01201120112011201 . . . i3 n~Ž~‘„Ž{•‘‚ ™‘† ‡~† Œ† ‘Ž‚†• ~Ž†…‰Œ| ‚|Š~† Ž„‘Œ|
13
r•ŒˆŒ€|•‘‚ ‘Œ •’~•†‡™ ‘Œ ‘Ž†~•†‡™ ‡~† ‘Œ •‚‡~‚‹~•†‡™ ~Šy•‘’€‰~ ‘Œ’ 150
n~Ž~‘„Ž{•‘‚ ™‘† ‡~†
‘~ ‘Ž|~ ~Š~•‘š€‰~‘~ ‚|Š~† •‚Ž†Œ•†‡y
X•‘– ~‡ŒˆŒ’…|~ (pn ) •‘Œ N ›•‘‚ Š~ †•”š‚† pn ≥ 2 €†~ ‡y…‚ n
f‚–ŽŒš‰‚ Œ•Œ†~•{•Œ‘‚ ~‡ŒˆŒ’…|~ (an ) •‘Œ Z ›•‘‚ Š~ †•”š‚† P
0 ≤ an ≤ pn − 1 €†~ ‡y…‚ n ‡~†
an
›•‘‚ Š~ ‰„Š †•”š‚† ‘‚ˆ†‡y an = pn − 1 _•Œ•‚|‹‘‚ ™‘† „ •‚†Žy +∞
n=1 p1 ···pn •’€‡ˆ|Š‚† ‡~† ™‘† ‘Œ
y…ŽŒ†•‰y ‘„• ~Š{‡‚† •‘Œ [0, 1)
_•Œ•‚|‹‘‚ ™‘† €†~ ‡y…‚ x ∈ [0, 1) ’•yŽ”‚† ‰ŒŠ~•†‡{ ~‡ŒˆŒ’…|~ (an ) •‘Œ Z ›•‘‚ Š~
P†•”š‚†a0n ≤
an ≤ pn − 1 €†~ ‡y…‚ n ›•‘‚ Š~ ‰„Š †•”š‚† ‘‚ˆ†‡y an = pn − 1 ‡~† ›•‘‚ x = +∞
n=1 p1 ···pn
`Ž‚|‘‚ ~Š~•ŽŒ‰†‡™ ‘š•Œ €†~ ‘„Š ~‡ŒˆŒ’…|~ (an )
hŽ†‘{Ž†~ •š€‡ˆ†•„• •‚†Ž›Š
P
hogqeogm qmr&$8&+<
e •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š €†~ ‡y…‚ ǫ > 0 ’•yŽ”‚†
P
n0 ›•‘‚ Š~ †•”š‚† nk=m+1 xk = |xm+1 + · · · + xn | < ǫ €†~ ‡y…‚ m, n ‰‚ n > m ≥ n0
qŒ y…ŽŒ†•‰~ xn0 pn0 + · · · + x1 p + x0 ŒŠŒ‰yƒ‚‘~† p ~•†‡™ ~Šy•‘’€‰~ ‘Œ’ x ‡~† •’‰•Œˆ|ƒ‚‘~† hxn0 . . . x1 x0 ip
m† x0 , x1 , . . . , xn0 ŒŠŒ‰yƒŒŠ‘~† p ~•†‡y •„“|~ ‘Œ’ x
a‚Š|‡‚’•„ ‘–Š p ~•†‡›Š ~Š~•‘’€‰y‘–Š
_•™•‚†‹„Pf‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn
e •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š „ ~‡ŒˆŒ’…|~ (sn ) •’€‡ˆ|Š‚† { †•Œ•šŠ~‰~ ~Š ‡~† ‰™ŠŒ
~Š „ (sn ) ‚|Š~† ~‡ŒˆŒ’…|~ &DXFK\ qŒ ™‘† „ (sn ) ‚|Š~† ~‡ŒˆŒ’…|~ &DXFK\ •„‰~|Š‚† ™‘† €†~ ‡y…‚
ǫ > 0 ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚†
|sn − sm | < ǫ
€†~ ‡y…‚ n, m ‰‚ n > m ≥ n0 e ~•™•‚†‹„ ‘‚ˆ‚†›Š‚† •†™‘† •~Ž~‘„ŽŒš‰‚ ™‘† †•”š‚†
|xm+1 + · · · + xn | = |(x1 + · · · + xn ) − (x1 + · · · + xm )| = |sn − sm |
€†~ ‡y…‚ n, m ‰‚ n > m
P
j‚Ž†‡z• “ŒŽz• •†~‘’•›ŠŒ’‰‚ ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ &DXFK\ –• ‚‹{• „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚†
~Š ‡~† ‰™ŠŒ ~Š
P
OLPm,n→+∞ nk=m+1 xk = 0.
P
1
n~Žy•‚†€‰~
f~ ‹~Š~•Œš‰‚ ‘„Š ~Ž‰ŒŠ†‡{ •‚†Žy +∞
n=1 n
P+∞ 1
X•‘– ™‘† „ •‚†Žy n=1 n •’€‡ˆ|Š‚† q™‘‚ ‰‚ ǫ = 21 ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚†
1
+ · · · + n1 | <
| m+1
1
2
€†~ ‡y…‚ m, n ‰‚ n > m ≥ n0 VŽ~ ‰‚ n = 2m •’Š‚•y€‚‘~† ™‘† †•”š‚†
1
+ ··· +
| m+1
1
2m |
1
2
<
€†~ ‡y…‚ m ≥ n0 [‰–•
1
+ ··· +
| m+1
1
2m |
=
1
m+1
+ ··· +
1
2m
≥
1
2m
+ ··· +
1
2m
=
2m−m
2m
=
1
2
‡~† ~•™ ‘†• •šŒ
‘‚ˆ‚’‘~|‚• •”z•‚†• ‡~‘~ˆ{€Œ’‰‚ •‚ y‘Œ•Œ
P+∞
VŽ~ „ •‚†Žy n=1 n1 ~•Œ‡ˆ|Š‚† Œ•™‘‚ –• •‚†Žy ‰„ ~ŽŠ„‘†‡›Š ™Ž–Š ~•Œ‡ˆ|Š‚† •‘Œ +∞
_•™ˆ’‘„ •š€‡ˆ†•„
P+∞
~•Œˆš‘–• ~Š „ •‚†Žy ‰‚ ‰„ ~ŽŠ„‘†‡Œš•
mogpjmp
iz‰‚
™‘†
„
•‚†Žy
n=1 xn •’€‡ˆ|Š‚†
P+∞
P+∞
™ŽŒ’•
n=1 |xn | < +∞
n=1 |xn | •’€‡ˆ|Š‚† { †•Œ•šŠ~‰~ ~Š
−
mogpjmp
a†~ ‡y…‚ x ŒŽ|ƒŒ’‰‚ x+ = |x|+x
2 ‡~† x =
−
‰zŽŒ• ‘Œ’ x ‡~† Œ x ŒŠŒ‰yƒ‚‘~† ‰„ …‚‘†‡™ ‰zŽŒ• ‘Œ’ x
|x|−x
2
m x+ ŒŠŒ‰yƒ‚‘~† ‰„ ~ŽŠ„‘†‡™
a†~ •~Žy•‚†€‰~ ‚|Š~† 3+ = 3 ‡~† 3− = 0 ‚|Š~† (−3)+ = 0 ‡~† (−3)− = 3 ‡~† ‚|Š~† 0+ = 0
‡~† 0− = 0
n~Ž~‘„Ž{•‘‚ ‘†• ~•ˆz• •”z•‚†•
x+ + x− = |x|,
x+ − x− = x,
0 ≤ x+ ≤ |x|, 0 ≤ x− ≤ |x|.
P
hogqeogm _nmirqep prahigpep _Š „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–• ‘™‘‚ •’€‡ˆ|
Š‚† ‡~†
P+∞ P+∞
n=1 xn ≤
n=1 |xn |.
P
nŽ›‘„ ~•™•‚†‹„ X•‘– ™‘† „ •‚†Žy +∞
n=1 |xn | •’€‡ˆ|Š‚† ‡~† z•‘– ǫ > 0
pš‰“–Š~ ‰‚ ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ &DXFK\ ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚† |xm+1 | + · · · + |xn | < ǫ ‡~†
‚•Œ‰zŠ–•
|xm+1 + · · · + xn | ≤ |xm+1 | + · · · + |xn | < ǫ
nŽŒ•z‹‘‚ ‘„Š Œ‰Œ†™‘„‘~ ~’‘{• ‘„• ~•™•‚†‹„• ‘„• ~•™‡ˆ†•„• ‘„• ~Ž‰ŒŠ†‡{• •‚†Žy• ‰‚ ‘„Š ~•™•‚†‹„ •‘Œ •~Žy•‚†€‰~
‰‚ ‘„ ˆš•„ ‘„• y•‡„•„•
‡~† ‡’Ž|–• ‰‚ ‘„ ˆš•„ ‘„• y•‡„•„•
€†~ ‡y…‚ m, n ‰‚ n > m ≥ n0
P
VŽ~ ‡~† •yˆ† •š‰“–Š~ ‰‚ ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ &DXFK\ „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚†
q›Ž~ ‚•‚†•{ †•”š‚† −|xn | ≤ xn ≤ |xn | €†~ ‡y…‚ n •’Š‚•y€‚‘~†
P+∞
P+∞
P
− +∞
n=1 |xn |
n=1 xn ≤
n=1 |xn | ≤
P+∞ P+∞
‡~† ‚•Œ‰zŠ–• n=1 xn ≤ n=1 |xn |
P
b‚š‘‚Ž„ ~•™•‚†‹„ X•‘– ™‘† „ •‚†Žy +∞
n=1 |xn | •’€‡ˆ|Š‚†
>~@
‡~†
~•™
‘†•
~Š†•™‘„‘‚•
€†~ ‘Œ’• xn •’Š‚•y€‚‘~† ™‘† ‡~† Œ† •‚†Žz•
_•™
‘„Š
•Ž™‘~•„
P+∞ +
P+∞ −
x
‡~†
x
•’€‡ˆ|ŠŒ’Š
n=1 n
n=1 n
P
c•‚†•{ †•”š‚† xn = xn + − xn − €†~ ‡y…‚ n •’Š‚•y€‚‘~† ™‘† ‡~† „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ‡~†
P+∞
P+∞
P
P
+∞
+∞
+
−
+
−
n=1 xn =
n=1 (xn − xn ) =
n=1 xn −
n=1 xn .
c•|•„•
P+∞ P+∞ + P+∞ − P+∞ + P+∞ − ≤
+
n=1 xn =
n=1 xn −
n=1 xn
n=1 xn
n=1 xn
P+∞ + P+∞ − P+∞
P
+∞
= n=1 xn + n=1 xn = n=1 (xn + + xn − ) = n=1
|xn |,
™•Œ’ •‘„Š ‘‚ˆ‚’‘~|~ †•™‘„‘~ ”Ž„•†‰Œ•Œ†{•~‰‚ ™‘† †•”š‚† |xn | = xn + + xn − €†~ ‡y…‚ n
P
P+∞
≤
_Š •Œš‰‚ ‘„Š ~Š†•™‘„‘~ +∞
x
n
n=1
n=1 |xn | –• €‚Š|‡‚’•„ ‘–Š |x1 + x2 | ≤ |x1 | + |x2 |
|x1 + x2 + x3 | ≤ |x1 | + |x2 | + |x3 | ‡ ‘ ˆ ‘™‘‚ •†‡~†ŒˆŒ€‚|‘~† Œ ™ŽŒ• ‘Ž†€–Š†‡{ ~Š†•™‘„‘~ €†~
‘„Š ~Š†•™‘„‘~ ~’‘{
P
(−1)n−1
n~Žy•‚†€‰~
e •‚†Žy +∞
•’€‡ˆ|Š‚† •†™‘† •’€‡ˆ|Š‚† ~•Œˆš‘–•
2
n=1
P+∞ (−1)n−1 nP+∞ 1
= n=1 n2 •’€‡ˆ|Š‚†
nŽy€‰~‘† „ •‚†Žy n=1
n2
n~Žy•‚†€‰~
b‚Š †•”š‚† ‘Œ ~Š‘|•‘ŽŒ“Œ ‘Œ’ ‡Ž†‘„Ž|Œ’ ~•™ˆ’‘„• •š€‡ˆ†•„•
P+∞ (−1)n−1 P
(−1)n−1
=
•’€‡ˆ|Š‚†
~ˆˆy
™”†
~•Œˆš‘–•
~“Œš
a†~ •~Žy•‚†€‰~ „ •‚†Žy +∞
n=1
n=1
n
n
P+∞ 1
n=1 n = +∞
P
nomq_pe
>~@ _Š †•”š‚† |xn | ≤ yn €†~ ‡y…‚ n ‡~† „ •‚†Žy +∞
yn •’€‡ˆ|Š‚†
‘™‘‚ „ •‚†Žy
n=1
P
P
P+∞
+∞
+∞
x
•’€‡ˆ|Š‚†
~•Œˆš‘–•
‡~†
‚•Œ‰zŠ–•
•’€‡ˆ|Š‚†
c•|•„•
‚|Š~†
x
≤
n=1 yn
n=1 n
n=1 n
>•@ X•‘– ™‘† †•”š‚† yn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† „ ~‡ŒˆŒ’…|~ |xynn | •’€‡ˆ|Š‚† { €‚Š†‡™‘‚Ž~
P+∞
P
‚|Š~† “Ž~€‰zŠ„ _Š „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~†
n=1 yn •’€‡ˆ|Š‚† ‘™‘‚ „ •‚†Žy
‚•Œ‰zŠ–• •’€‡ˆ|Š‚†
P
P+∞
_•™•‚†‹„ >~@ _Š P
„ •‚†Žy +∞
•’Š‚•y€‚‘~†
™‘† „ •‚†Žy
n=1 yn •’€‡ˆ|Š‚†
n=1 |xn | •’€‡ˆ|Š‚†
P
P
P
+∞ xn ≤ +∞ |xn | ≤ +∞ yn
Œ•™‘‚ ‡~† „ •‚†Žy +∞
x
•’€‡ˆ|Š‚†
‡~†
n
n=1
n=1
n=1
n=1
>•@ V‰‚•„ •’Šz•‚†~ ‘„• •Ž™‘~•„•
‡~† ‘Œ’ ‡Ž†‘„Ž|Œ’ ~•™ˆ’‘„• •š€‡ˆ†•„•
b†y“ŒŽ~ •”™ˆ†~ P
•Œ’ ‚|”~Š €|Š‚† ‰‚‘y ~•™ ‘„ŠP
•Ž™‘~•„
z”Œ’Š …z•„ ‡~† ‚•› ‰™ŠŒ •Œ’ ‚“~Ž
+∞
x
b„ˆ~•{
‘~ |•†~ •”™ˆ†~ •†~‘’•›ŠŒŠ‘~†
|x
|
~Š‘|
‘„•
•‚†Žy•
‰™ƒŒŠ‘~† •‘„ •‚†Žy +∞
n
n
n=1
n=1
P
•‚ •”z•„ ‰‚ ‘„Š ~•™ˆ’‘„ •š€‡ˆ†•„ ‘„• +∞
x
n=1 n
P
(−2)n
n~Žy•‚†€‰~
a†~ Š~ ‰‚ˆ‚‘{•Œ’‰‚ ‘„ •‚†Žy +∞
n=1 3n +2n €Žy“Œ’‰‚
(−2)n n n =
3 +2
Œ•™‘‚
c•‚†•{ „ •‚†Žy
•‘~ ~•Œˆš‘–•
2n
3n +2n
1
= ( 32 )n 1+(2/3)
n,
n
2 n
| 3(−2)
n +2n |/( 3 ) → 1.
P+∞
2 n
n=1 ( 3 )
•’€‡ˆ|Š‚† •’Š‚•y€‚‘~† ™‘† „ •‚†Žy
(−2)n
n=1 3n +2n
P+∞
•’€‡ˆ|Š‚† ‡~† ‰yˆ†
f~ •~Ž~‘„Ž{•‚‘‚ ™‘† ‘~ ‚•™‰‚Š~ •šŒ ‡Ž†‘{Ž†~ •Œ’ …~ ‰‚ˆ‚‘{•Œ’‰‚ ‚“~Ž‰™ƒŒŠ‘~† Œ’•†~
•‘†‡y •‚ •‚†Žz• ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• ™‘~Š •y•‚† ~’‘›Š ‘–Š ‡Ž†‘„Ž|–Š •ŽŒ‡š•‘‚† …‚‘†‡™
•’‰•zŽ~•‰~ ~’‘™ ‚|Š~† „ ~•™ˆ’‘„ •š€‡ˆ†•„ ‰†~• •‚†Žy• j‚ yˆˆ~ ˆ™€†~ ~Š ~Š‘†‰‚‘–•|ƒŒ’‰‚ ‰†~
•‚†Žy „ Œ•Œ|~ •’€‡ˆ|Š‚† ~ˆˆy •‚Š •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† •‚Š ‘Œ €Š–Ž|ƒŒ’‰‚ ‘™‘‚ ‘~ ‡Ž†‘{Ž†~
~’‘y •‚Š …~ •›•Œ’Š …‚‘†‡™ •’‰•zŽ~•‰~ X‘•† …~ ‡~‘~ˆy•Œ’‰‚ ™‘† „ •‚†Žy •‚Š •’€‡ˆ|Š‚† ~•Œˆš
‘–• Œ•™‘‚ ~•Œ‰zŠ‚† Š~ ‚‹‚‘y•Œ’‰‚ ‰‚ ‘~ ‡Ž†‘{Ž†~ •Œ’ ~‡ŒˆŒ’…ŒšŠ ~Š „ •‚†Žy •’€‡ˆ|Š‚† ’•™
•’Š…{‡„
hogqeogm imamr qmr ' $/(0%(57 X•‘– xn 6= 0 €†~ ‡y…‚ n
P
< 1 ‘™‘‚ „ •‚†Žy +∞ xn •’€‡ˆ|Š‚† ~•Œˆš‘–•
(i) _Š OLP xxn+1
n=1
n
xn+1 P+∞
(ii) _Š OLP xn > 1 ‘™‘‚ „ •‚†Žy n=1 xn ~•Œ‡ˆ|Š‚†
P
≤ 1 ≤ OLP xn+1 ‘™‘‚ •‚Š ’•yŽ”‚† €‚Š†‡™ •’‰•zŽ~•‰~ €†~ ‘„ •‚†Žy +∞ xn
(iii) _Š OLP xxn+1
n=1
xn
n
xn+1 _•™•‚†‹„ (i) f‚–ŽŒš‰‚ zŠ~Š Œ•Œ†ŒŠ•{•Œ‘‚ a ›•‘‚ OLP xn < a < 1
≤ a €†~ ‡y…‚ n ≥ n0 q™‘‚ €†~ ‡y…‚ n ≥ n0 + 1 †•”š‚†
q™‘‚ ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚† xxn+1
n
xn +1 n xn−1 0 |xn | ≤ aa · · · a|xn | = an−n0 |xn | = |xnn0 | an = can ,
|xn | = xxn−1
0
0
0
xn−2 · · · xn
a 0
0
P
P
|x |
™•Œ’ c = ann00 c•‚†•{ 0 ≤ a < 1 „ •‚†Žy +∞
an •’€‡ˆ|Š‚† Œ•™‘‚ ‡~† „ +∞
n=n
+1
n=n0 +1 |xn |
0
P+∞
•’€‡ˆ|Š‚† VŽ~ „ •‚†Žy n=1 |xn | •’€‡ˆ|Š‚†
≥ 1 €†~ ‡y…‚ n ≥ n0 b„ˆ~•{ †•”š‚†
(ii) r•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚† xxn+1
n
|xn | ≥ |xn−1 | ≥ · · · ≥ |xn0 | > 0
P
€†~ ‡y…‚ n ≥ n0 + 1 VŽ~ •‚Š †•”š‚† xn → 0 ‡~† ‚•Œ‰zŠ–• „ •‚†Žy +∞
n=1 xn •‚Š •’€‡ˆ|Š‚†
1/(n+1)2 1/(n+1) P+∞ 1
P+∞ 1
(iii) a†~ ‘†• •‚†Žz• n=1 n ‡~† n=1 n2 ‚|Š~† 1/n → 1 ‡~† 1/n2 → 1 + •Ž›‘„ •‚†Žy
~•Œ‡ˆ|Š‚† ‡~† „ •‚š‘‚Ž„ •’€‡ˆ|Š‚†
p
P
hogqeogm ogd_p qmr &$8&+< (i) _Š OLP n |xn | < 1 ‘™‘‚ „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚†
~•Œˆš‘–•
p
P
(ii) _Š OLP n |xn | > 1 ‘™‘‚ „ •‚†Žy +∞
n=1 xn ~•Œ‡ˆ|Š‚†
p
P
n
(iii) _Š OLP |xn | = 1 ‘™‘‚ •‚Š ’•yŽ”‚† €‚Š†‡™ •’‰•zŽ~•‰~ €†~ ‘„ •‚†Žy +∞
n=1 xn
p
_•™•‚†‹„ (i) f‚–ŽŒš‰‚ zŠ~Š Œ•Œ†ŒŠ•{•Œ‘‚
a ›•‘‚ OLP n |xn | < a < 1
p
n
q™‘‚ ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚† |xn | ≤ a ‡~† ‚•Œ‰zŠ–•
|xn | ≤ an
P
P
€†~ ‡y…‚ n ≥ n0 c•‚†•{ 0 ≤ a < 1 „ •‚†Žy +∞
an •’€‡ˆ|Š‚† Œ•™‘‚ ‡~† „ •‚†Žy +∞
n=n
n=n0 |xn |
0
P+∞
•’€‡ˆ|Š‚† VŽ~ „ •‚†Žy n=1 |xn | •’€‡ˆ|Š‚†
p
(ii) g•”š‚† n |xn | ≥ 1 ‡~† ‚•Œ‰zŠ–•
|xn | ≥ 1
P
•’€‡ˆ|Š‚†
€†~ y•‚†ŽŒ’• n VŽ~ •‚Š †•”š‚† xn → 0 Œ•™‘‚ „ •‚†Žy +∞
n=1 xn •‚Š
p
p
P+∞ 1
P+∞ 1
n
n
(iii) a†~ ‘†• •‚†Žz• n=1 n ‡~† n=1 n2 ‚|Š~† |1/n| → 1 ‡~† |1/n2 | → 1 e •Ž›‘„ •‚†Žy
~•Œ‡ˆ|Š‚† ‚Š› „ •‚š‘‚Ž„ •’€‡ˆ|Š‚†
P
p‘„Š ‚“~Ž‰Œ€{ ‘–Š ‡Ž†‘„Ž|–Š ˆ™€Œ’ ‡~† Ž|ƒ~• •‚ •’€‡‚‡Ž†‰zŠ‚• •‚†Žz• +∞
n=1 xn ‘†• •‚Ž†•
•™‘‚Ž‚• “ŒŽz• ’•yŽ”Œ’Š ‘~ ™Ž†~
p
,
OLPn→+∞ xxn+1
OLPn→+∞ n |xn |.
n
q™‘‚ ™•–• €Š–Ž|ƒŒ’‰‚ ‡~† ‘Œ ”Ž„•†‰Œ•Œ†{•~‰‚ •‘†• ~•Œ•‚|‹‚†• ‘–Š ‰‚Ž›Š (iii) ‘–Š •šŒ ‡Ž†‘„
Ž|–Š ‚|Š~† OLP = OLP = OLP
•„ˆ~•{ ‘~ ‡Ž†‘{Ž†~ ‘Œ’ 'LULFKOHW ‘Œ’ $EHO ‡~† ‘–Š ‚Š~ˆˆ~••™‰‚Š–Š •ŽŒ•{‰–Š
P
an
n~Žy•‚†€‰~
f‚–ŽŒš‰‚ ‘„ •‚†Žy +∞
n=1 n
f~ ‚“~Ž‰™•Œ’‰‚ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ _Š a = 0 „ •‚†Žy •ŽŒ“~Š›• •’€‡ˆ|Š‚† ~•Œˆš‘–• _Š a 6= 0
‚|Š~†
an+1 /(n+1) = |a| n → |a|.
n
a /n
n+1
VŽ~ ~Š 0 < |a| < 1 „ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† ~Š |a| > 1 „ •‚†Žy ~•Œ‡ˆ|Š‚†
q›Ž~ …~ ‚“~Ž‰™•Œ’‰‚ ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~• c|Š~†
p
√
n
|an /n| = |a|/ n n → |a|.
c•Œ‰zŠ–• ~Š |a| < 1 „ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† ~Š |a| > 1 „ •‚†Žy ~•Œ‡ˆ|Š‚†
_Š |a| = 1 ‡~ŠzŠ~ ~•™ ‘~ •šŒ ‡Ž†‘{Ž†~ •‚Š •|Š‚† •’‰•zŽ~•‰~ Œ•™‘‚ ‚‹‚‘yƒŒ’‰‚ ‘†• •‚Ž†•‘›•‚†•
a = ±1 ~Š‚‹yŽ‘„‘~ ~•™ ‘~
P•šŒ ‡Ž†‘{Ž†~
1
_Š a = 1 „ •‚†Žy ‚|Š~† „ +∞
n=1 n „ Œ•Œ|~ ~•Œ‡ˆ|Š‚†
P
(−1)n
_Š a = −1 „ •‚†Žy ‚|Š~† „ +∞
„ Œ•Œ|~ •’€‡ˆ|Š‚† ~ˆˆy ™”† ~•Œˆš‘–•
n=1
n
P+∞ an
p’ŠŒˆ†‡y „ •‚†Žy n=1 n •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š −1 ≤ a < 1 ‡~† •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š
‡~† ‰™ŠŒ ~Š −1 < a < 1
P
an
n~Žy•‚†€‰~
f‚–ŽŒš‰‚ ‘„ •‚†Žy +∞
n=1 n2
f~ ‚“~Ž‰™•Œ’‰‚ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ _Š a = 0 „ •‚†Žy •ŽŒ“~Š›• •’€‡ˆ|Š‚† ~•Œˆš‘–• _Š a 6= 0
‚|Š~†
an+1 /(n+1)2 = |a| n2 2 → |a|.
an /n2
(n+1)
VŽ~ ~Š 0 < |a| < 1 „ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† ~Š |a| > 1 „ •‚†Žy ~•Œ‡ˆ|Š‚†
q›Ž~ …~ ‚“~Ž‰™•Œ’‰‚ ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~• c|Š~†
p
√
n
|an /n2 | = |a|/( n n)2 → |a|.
c•Œ‰zŠ–• ~Š |a| < 1 „ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† ~Š |a| > 1 „ •‚†Žy ~•Œ‡ˆ|Š‚†
_Š |a| = 1 ‡~ŠzŠ~ ~•™ ‘~P
•šŒ ‡Ž†‘{Ž†~ •‚Š •|Š‚† •’‰•zŽ~•‰~
1
_Š a = 1 „ •‚†Žy ‚|Š~† „ +∞
n=1 n2 „ Œ•Œ|~ •’€‡ˆ|Š‚† ~•Œˆš‘–•
P
(−1)n
„ Œ•Œ|~ ‚•|•„• •’€‡ˆ|Š‚† ~•Œˆš‘–•
_Š a = −1 „ •‚†Žy ‚|Š~† „ +∞
n=1
n2
P+∞ an
VŽ~ „ •‚†Žy n=1 n2 •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š |a| ≤ 1 ‡~† ~•Œ‡ˆ|Š‚† ~Š |a| > 1
P
an
n~Žy•‚†€‰~
p‘„ •‚†Žy +∞
n=0 n! ‚“~Ž‰™ƒŒ’‰‚ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’
_Š a = 0 „ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• _Š a 6= 0 ‚|Š~†
an+1 /(n+1)! = |a| → 0 < 1.
n+1
an /n!
VŽ~ „ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• €†~ ‡y…‚ a
e ‚“~Ž‰Œ€{ ‘Œ’ ‡Ž†‘„Ž|Œ’ Ž|ƒ~• ‚|Š~† •†Œ •š•‡Œˆ„ c|Š~†
p
√
n
|an /n!| = |a|/ n n!
‡~† ”Ž‚†~ƒ™‰~•‘‚ ‘Œ ™Ž†Œ
√
n
n! → +∞.
_Š Œ n ‚|Š~† yŽ‘†Œ•
n/2 .
n! = 1 · · · n2 ( n2 + 1) · · · n ≥ ( n2 + 1) · · · n ≥ ( n2 + 1)n/2 ≥ ( n+1
2 )
_Š Œ n ‚|Š~† •‚Ž†‘‘™•
n+1
n! = 1 · · · n−1
2
2 ···n ≥
n+1
2
j†~ ~•™•‚†‹„ €†~ ‘Œ ™Ž†Œ ~’‘™ ‚|Š~† •‘„Š y•‡„•„
(n+1)/2 ≥ ( n+1 )n/2 .
· · · n ≥ ( n+1
2 )
2
c•› z”Œ’‰‚ ‰†~ ~•ˆŒš•‘‚Ž„ ~•™•‚†‹„
√
1/2 €†~ ‡y…‚ n Œ•™‘‚ n n! → +∞ c•Œ‰zŠ–•
n! ≥ ( n+1
2 )
p
n
|an /n!| → 0 < 1,
P
an
Œ•™‘‚ „ •‚†Žy +∞
n=0
n! •’€‡ˆ|Š‚† ~•Œˆš‘–•
P+∞ an
e •‚†Žy n=0 n! ‚|Š~† †•†~†‘zŽ–• •„‰~Š‘†‡{ ‡~† …~ ‘{Š ‹~Š~•Œš‰‚ •‘Œ •~Žy•‚†€‰~
‡~†
‡~†
p‘„Š •‚Ž|•‘–•„ a = 1 €Š–Ž|ƒŒ’‰‚ ™‘† ‘Œ y…ŽŒ†•‰~
‡’Ž|–• •‘~ •~Ž~•‚|€‰~‘~
‘„• •‚†Žy• ‚|Š~† Œ ~Ž†…‰™• e
VŽ~ †•”š‚†
√
n
Z•–• ~Š~Ž–‘„…‚| ‡~Š‚|• ~Š ’•yŽ”‚† ‡y•Œ†~ •”z•„ ~Šy‰‚•~ •‘~ ‡Ž†‘{Ž†~ ˆ™€Œ’ ‡~† Ž|ƒ~•
•‘„Š
P+∞•‚Ž|•‘–•„ •Œ’ ‰•ŒŽŒšŠ Š~ ‚“~Ž‰Œ•…ŒšŠ ‡~† ‘~ •šŒ ‘~’‘Œ”Ž™Š–• •„ˆ~•{ ~Š z”Œ’‰‚ •‚†Žy
n=1 xn €†~ ‘„Š Œ•Œ|~ †•”š‚† xn 6= 0 €†~ ‡y…‚ n e ~•yŠ‘„•„ ‚|Š~† ™‘† ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~• ‚|Š~†
†•”’Ž™‘‚ŽŒ ~•™ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ b„ˆ~•{ ~Š ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ •|Š‚† ‡y•Œ†Œ ~•Œ‘zˆ‚•‰~ €†~
‘„ •š€‡ˆ†•„ ‘„• •‚†Žy• ‘™‘‚ ‡~† ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~• •|Š‚† ‘Œ |•†Œ ~•Œ‘zˆ‚•‰~ ‚Š› ’•yŽ”Œ’Š •~
Ž~•‚|€‰~‘~ •‚†Ž›Š €†~ ‘~ Œ•Œ|~ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ •‚Š •|Š‚† ~•Œ‘zˆ‚•‰~ ‚Š› ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~•
•|Š‚† [‰–• ‰‚Ž†‡z• “ŒŽz• ™•–• •‘Œ ‘‚ˆ‚’‘~|Œ •~Žy•‚†€‰~ ‚|Š~† •†Œ ‚š‡ŒˆŒ Š~ ‚“~Ž‰Œ•…‚| ‘Œ
‡Ž†‘{Ž†Œ ˆ™€Œ’ ~•™ ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~•
r•™ •’Š…{‡„ •š€‡ˆ†•„
mogpjmp
Š‚† ~•Œˆš‘–•
iz‰‚ ™‘† „ •‚†Žy
P+∞
n=1 xn
•’€‡ˆ|Š‚† ’•™ •’Š…{‡„ ~Š •’€‡ˆ|Š‚† ~ˆˆy •‚Š •’€‡ˆ|
~Š~“zŽ~‰‚ ™‘† „ •‚†Žy
n~Žy•‚†€‰~
i|€Œ •Ž†Š •‘Œ •~Žy•‚†€‰~
€‡ˆ|Š‚† ~ˆˆy ™”† ~•Œˆš‘–• b„ˆ~•{ „ •‚†Žy ~’‘{ •’€‡ˆ|Š‚† ’•™ •’Š…{‡„
P+∞
n=1
(−1)n−1
n
•’
p‘Œ •„‰‚|Œ ~’‘™ …~ ‡yŠŒ’‰‚ ‡y•Œ†~ •”™ˆ†~ €†~ ‘„Š zŠŠŒ†~ ‘„• •š€‡ˆ†•„• ‰†~• •‚†Žy•
nŽ›‘ŒŠ z•‘– ‰†~ •‚†Žy ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• qŒ Š~ •’€‡ˆ|Š‚† „ •‚†Žy ‚|Š~† †•Œ•šŠ~‰Œ ‰‚ ‘Œ
Š~ ‚|Š~† “Ž~€‰zŠ~ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘y ‘„• _’‘y ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ •„‰†Œ’Ž€ŒšŠ‘~† ‰‚ •†~
•Œ”†‡{ y…ŽŒ†•„ ‘–Š ™Ž–Š ‘„• •‚†Žy• Œ•™‘‚ ‚|Š~† “~Š‚Ž™ ™‘† €†~ Š~ ‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~
“Ž~€‰zŠ~ •Žz•‚† ‘Œ ‰z€‚…Œ• ‘–Š ™Ž–Š •ŽŒ•…‚‘z–Š Š~ ‚|Š~† ~Ž‡‚‘y ‰†‡Ž™ ™‘~Š •ŽŒ•…z‘Œ’‰‚
‰‚€yˆŒ’• ~Ž†…‰Œš• •Ž|•‡Œ’‰‚ ‰‚€yˆ~ ~…ŽŒ|•‰~‘~ ‚Š› ™‘~Š •ŽŒ•…z‘Œ’‰‚ ‰†‡ŽŒš• ~Ž†…‰Œš• •Ž|
„ Œ•Œ|~ ˆz‚† ™‘† ~Š ‰†~ •‚†Žy
•‡Œ’‰‚ ‰†‡Žy ~…ŽŒ|•‰~‘~ _’‘™ “~|Š‚‘~† ‡~† ~•™ ‘„Š •Ž™‘~•„
•’€‡ˆ|Š‚† ‘™‘‚ Œ† ™ŽŒ† ‘„• ‘‚|ŠŒ’Š •‘ŒŠ 0 [‰–• „ ~•ˆ{ •š€‡ˆ†•„ ‘–Š ™Ž–Š •‘ŒŠP0 •‚Š ~Ž‡‚|
1
~•™ ‰™Š„ ‘„• Š~ ‡yŠ‚† ‘„ •‚†Žy Š~ •’€‡ˆ|Š‚† a†~ •~Žy•‚†€‰~ ‡~† •‘†• •šŒ •‚†Žz• +∞
n=1 n2 ‡~†
P+∞ 1
n=1 n Œ† ™ŽŒ† ‘‚|ŠŒ’Š •‘ŒŠ 0 ~ˆˆy „ •Ž›‘„ •’€‡ˆ|Š‚† ‚Š› „ •‚š‘‚Ž„ •‚Š •’€‡ˆ|Š‚† n~Ž~‘„Ž{
•‘‚ ™‘† Œ† ™ŽŒ† ‘„• •Ž›‘„• •‚†Žy• ‚|Š~† •Œˆš ‰†‡Ž™‘‚ŽŒ† ~•™ ‘Œ’• ~Š‘|•‘Œ†”Œ’• ™ŽŒ’• ‘„• •‚š‘‚Ž„•
2
•‚†Žy• nŽy€‰~‘† 1/n
1/n → 0 b„ˆ~•{ ‘Œ ‰z€‚…Œ• ‘–Š ™Ž–Š ‘„• •Ž›‘„• •‚†Žy• ‚|Š~† ‘™•Œ ‰†‡Ž™
›•‘‚ „ •‚†Žy •’€‡ˆ|Š‚† ‚Š› ‘Œ ‰z€‚…Œ• ‘–Š ™Ž–Š ‘„• •‚š‘‚Ž„• •‚†Žy• •‚Š ‚|Š~† ‘™•Œ ‰†‡Ž™ ™•Œ
…~ z•Ž‚•‚ €†~ Š~ •’€‡ˆ|Š‚† ‡~† ~’‘{ _’‘™ ‘Œ •~†”Š|•† ‰‚ ‘Œ ‰z€‚…Œ• ‘–Š ™Ž–Š “~|Š‚‘~† ‡~…~Žy
‡~† •‘„Š •Ž™‘~•„
qŒ •~•†‡™ ‘„• •’‰•zŽ~•‰~ ‚|Š~† ™‘† ~Š ‰†~ •‚†Žy ‰‚ ‰‚€~ˆš‘‚ŽŒ’• ™ŽŒ’•
•’€‡ˆ|Š‚† ‘™‘‚ ‡~† „ •‚†Žy ‰‚ ‘Œ’• ‰†‡Ž™‘‚ŽŒ’• ™ŽŒ’• •’€‡ˆ|Š‚†
[ˆ~ ‘~ •ŽŒ„€Œš‰‚Š~ z”Œ’Š –• •~•†‡{ •ŽŒ˜•™…‚•„ ™‘† ~Š~“‚Ž™‰~•‘‚ •‚ •‚†Žz• ‰‚ ‰„ ~ŽŠ„‘†‡Œš•
™ŽŒ’• zŠ~• ‰„ ~ŽŠ„‘†‡™• ~Ž†…‰™• ‘~’‘|ƒ‚‘~† ‰‚ ‘Œ ‰z€‚…™• ‘Œ’
e ‡~‘y•‘~•„ ~ˆˆyƒ‚† ‡y•–• ™‘~Š ‚Ž€~ƒ™‰~•‘‚ ‰‚ •‚†Žz• ‘–Š Œ•Œ|–Š Œ† ™ŽŒ† z”Œ’Š ‰‚‘~•~ˆˆ™
‰‚ŠŒ •Ž™•„‰Œ h~† •yˆ† €†~ Š~ •’€‡ˆ|Š‚† ‰†~ •‚†Žy •Žz•‚† Œ† ™ŽŒ† ‘„• Š~ ‘‚|ŠŒ’Š •‘ŒŠ 0 ‡~†
‚•Œ‰zŠ–• ‘Œ ‰z€‚…™• ‘Œ’• •’Š‚”|ƒ‚† Š~ •~|ƒ‚† Ž™ˆŒ [‰–• ‘Œ ‰z€‚…Œ• ‘–Š ™Ž–Š •‚Š •~|ƒ‚† •†~
P
P
(−1)n−1
1
‡~† +∞
‘ŒŠ ‡~…ŒŽ†•‘†‡™ Ž™ˆŒ b‚|‘‚ €†~ •~Žy•‚†€‰~ ‘†• •‚†Žz• +∞
n=1 n m† ™ŽŒ† ‘Œ’•
n=1
n
z”Œ’Š ~‡Ž†•›• ‘Œ |•†Œ ‰z€‚…Œ• [‰–• ‚Š› ‘Œ ‰z€‚…Œ• ~’‘™ •‚Š ‚|Š~† ~Ž‡‚‘y ‰†‡Ž™ ›•‘‚ Š~ •’
€‡ˆ|Š‚† „ •‚š‘‚Ž„ •‚†Žy ‚|Š~† ~Ž‡‚‘y ‰†‡Ž™ ›•‘‚ Š~ •’€‡ˆ|Š‚† „ •Ž›‘„ •‚†Žy m ˆ™€Œ• ‚|Š~† ™‘†
~•™ ‘~ •†~“ŒŽ‚‘†‡y •Ž™•„‰~ •ŽŒ‡~ˆ‚|‘~† ~ˆˆ„ˆŒ~Š~|Ž‚•„ ‘–Š ™Ž–Š ‡~‘y ‘„Š y…ŽŒ†•{ ‘Œ’• ‡~†
z‘•† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ •~Ž~‰zŠŒ’Š ’•™ zˆ‚€”Œ _’‘™ ~‡Ž†•›• ‘Œ “~†Š™‰‚ŠŒ •~Ž~‘„Ž‚|‘~†
b‚|‘‚ ‘„Š y•‡„•„
•‚ Œ•Œ†~•{•Œ‘‚ •‚†Žy •Œ’ •’€‡ˆ|Š‚† ’•™ •’Š…{‡„ qŒ ‰z€‚…Œ• ‘–Š ™Ž–Š ‘„• •‚Š ‚|Š~† ~Ž‡‚‘y
‰†‡Ž™ ›•‘‚ Š~ •’€‡ˆ|Š‚† „ •‚†Žy ~•Œˆš‘–• •„ˆ~•{ Š~ •’€‡ˆ|Š‚† „ •‚†Žy ‘–Š ‰‚€‚…›Š ‘–Š ™Ž–Š
~ˆˆy ‚|Š~† ~Ž‡‚‘y ‰†‡Ž™ ›•‘‚ ‰‚‘y ‡~† ~•™ ‘†• ~ˆˆ„ˆŒ~Š~†Žz•‚†• ˆ™€– •†~“ŒŽ‚‘†‡›Š •ŽŒ•{‰–Š
„ •‚†Žy Š~ •’€‡ˆ|Š‚† X‘•† ‘Œ ‡Ž†‘{Ž†Œ ~•™ˆ’‘„• •š€‡ˆ†•„• “~|Š‚‘~† ˆŒ€†‡™ ~Š ‘Œ ‰z€‚…Œ• ‘–Š
™Ž–Š ‰†~• •‚†Žy• ‚|Š~† ~Ž‡‚‘y ‰†‡Ž™ ›•‘‚ „ •‚†Žy ‘–Š ‰‚€‚…›Š ~’‘›Š Š~ •’€‡ˆ|Š‚† ‘™‘‚ ‚|Š~†
~Ž‡‚‘y ‰†‡Ž™ ›•‘‚ ‰‚‘y ‡~† ~•™ ‘†• ~ˆˆ„ˆŒ~Š~†Žz•‚†• ˆ™€– •†~“ŒŽ‚‘†‡›Š •ŽŒ•{‰–Š „ •‚†Žy
‘–Š |•†–Š ‘–Š ™Ž–Š Š~ •’€‡ˆ|Š‚†
i™€– ~’‘{• ‘„• •†~“ŒŽy• ~Šy‰‚•~ •‘„ “š•„ ‘„• •š€‡ˆ†•„• ‘–Š •‚†Ž›Š ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’•
‡~† •‘„ “š•„ ‘„• •š€‡ˆ†•„• ‘–Š •‚†Ž›Š ‰‚ €‚Š†‡Œš• ™ŽŒ’• ’•yŽ”‚† ‡~† ~Š‘|•‘Œ†”„ •†~“ŒŽy
~Šy‰‚•~ •‘†• ”Ž„•†‰Œ•Œ†Œš‰‚Š‚• ‰‚…™•Œ’• ‰‚ˆz‘„• ‘„• •š€‡ˆ†•{• ‘Œ’• a†~ •~Žy•‚†€‰~ „ •š
‡~† •‘~ ~Š‘|
€‡Ž†•„ ~Š‘|•‘Œ†”–Š ™Ž–Š ™•–• ~’‘{ ‚‡“Žyƒ‚‘~† •‘~ •šŒ ‰zŽ„ ‘„• •Ž™‘~•„•
•‘Œ†”~ ‰zŽ„ ‘„• •Ž™‘~•„•
•‚Š ‚“~Ž‰™ƒ‚‘~† •‚ •‚†Žz• •Œ’ •’€‡ˆ|ŠŒ’Š ’•™ •’Š…{‡„ ~‡Ž†•›•
‚•‚†•{ •Ž™‡‚†‘~† €†~ •š€‡Ž†•„ ‘–Š ‰‚€‚…›Š ‘–Š ~Š‘|•‘Œ†”–Š ™Ž–Š e ‰z…Œ•Œ• ‘„• •š€‡Ž†•„•
‚“~Ž‰™ƒ‚‘~† ‰™ŠŒ €†~ ‘„ ‰‚ˆz‘„ ‘„• •š€‡ˆ†•„• •‚†Ž›Š ‰„ ~ŽŠ„‘†‡›Š ™Ž–Š { ‘„• ~•™ˆ’‘„• •š
€‡ˆ†•„• •‚†Ž›Š •„ˆ~•{ ‡~† •yˆ† ‘„• •š€‡ˆ†•„• •‚†Ž›Š ‰„ ~ŽŠ„‘†‡›Š ™Ž–Š m† ‰z…Œ•Œ† ‰‚ˆz
‘„• ‘„• •š€‡ˆ†•„• •‚†Ž›Š Œ† Œ•Œ|‚• •‚Š •’€‡ˆ|ŠŒ’Š ~•Œˆš‘–• ‚|Š~† ~’‘z• •Œ’ …~ •Œš‰‚ ‘›Ž~
‘~ ‡Ž†‘{Ž†~ ‘Œ’ 'LULFKOHW ‡~† ‘Œ’ $EHO ‡~† –• •™Ž†•‰~ ‘Œ ‡Ž†‘{Ž†Œ ‚Š~ˆˆ~••™‰‚Š–Š •ŽŒ•{‰–Š
qrnmp _fomgpep qmr $%(/ X•‘– ~‡ŒˆŒ’…|‚• (an ) (bn ) ‡~† sn = a1 +· · ·+an ‘~ ‰‚Ž†‡y
~…ŽŒ|•‰~‘~ ‘„• •Ž›‘„• a†~ ‡y…‚ n, m ‰‚ n > m †•”š‚†
Pn
Pn
k=m+1 ak bk =
k=m+1 sk (bk − bk+1 ) + sn bn+1 − sm bm+1 .
_•™•‚†‹„ c|Š~†
Pn
k=m+1 ak bk
=
=
=
=
Pn
Pn
k=m+1 sk−1 bk
k=m+1 sk bk −
k=m+1 (sk − sk−1 )bk =
Pn−1
Pn
sk bk − k=m sk bk+1
Pn
Pnk=m+1
k=m+1 sk bk+1 + sn bn+1 − sm bm+1
k=m+1 sk bk −
Pn
k=m+1 sk (bk − bk+1 ) + sn bn+1 − sm bm+1 .
Pn
p‘ŒŠ ‘š•ŒP
y…ŽŒ†•„• ‘Œ’ $EHOP
•~Ž~‘„Ž{•‘‚ ‘ŒŠ ‘Ž™•Œ ‰‚ ‘ŒŠ Œ•Œ|Œ €|Š‚‘~† „ ‰‚‘y•~•„ ~•™
‘Œ y…ŽŒ†•‰~ nk=m+1 ak bk •‘Œ nk=m+1 sk (bk − bk+1 ) + sn bn+1 − sm bm+1 q„ …z•„ ‘–Š ak
•~|ŽŠŒ’Š ‘~ •†~•Œ”†‡y ‰‚Ž†‡y ~…ŽŒ|•‰~‘y ‘Œ’• sk ‡~† ‘„ …z•„ ‘–Š bk •~|ŽŠŒ’Š Œ† •†~•Œ”†‡z•
•†~“ŒŽz• ‘Œ’• bk − bk+1 c‰“~Š|ƒŒŠ‘~† ‡~† Œ† ~‡Ž~|Œ† ™ŽŒ† sn bn+1 ‡~† sm bm+1
hogqeogm qmr ',5,&+/(7 X•‘– ~‡ŒˆŒ’…|‚• (an ) (bn ) ‡~† sn = a1 + · · · + an ‘~ ‰‚Ž†‡y
~…ŽŒ|•‰~‘~
P ‘„• •Ž›‘„• _Š „ (bn ) ‚|Š~† “…|ŠŒ’•~ ~Š bn → 0 ‡~† ~Š „ (sn ) ‚|Š~† “Ž~€‰zŠ„ ‘™‘‚ „
•‚†Žy +∞
n=1 an bn •’€‡ˆ|Š‚†
_•™•‚†‹„ r•yŽ”‚† M ›•‘‚ Š~ †•”š‚† |sn | ≤ M €†~ ‡y…‚ n c•|•„• ‚•‚†•{ „ (bn ) ‚|Š~† “…|ŠŒ’•~
‡~† z”‚† ™Ž†Œ 0 †•”š‚† bn ≥ 0 €†~ ‡y…‚ n
X•‘– ǫ > 0 q™‘‚ ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚† bn ≤ 2Mǫ+1 €†~ ‡y…‚ n ≥ n0 `y•‚† ‘Œ’ ‘š•Œ’
y…ŽŒ†•„• ‘Œ’ $EHO €†~ ‡y…‚ n, m ‰‚ n > m ≥ n0 †•”š‚†
Pn
Pn
k=m+1 sk (bk − bk+1 ) + sn bn+1 − sm bm+1
k=m+1 ak bk =
Pn
≤ k=m+1 |sk |(bk − bk+1 ) + |sn |bn+1 + |sm |bm+1
P
≤ M nk=m+1 (bk − bk+1 ) + M bn+1 + M bm+1
= M (bm+1 − bn+1 ) + M bn+1 + M bm+1 = 2M bm+1 ≤
_•™ ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ &DXFK\ •’Š‚•y€‚‘~† ™‘† „ •‚†Žy
P+∞
n=1 an bn
•’€‡ˆ|Š‚†
2M ǫ
2M +1
< ǫ.
hogqeogm qmr $%(/ X•‘– ~‡ŒˆŒ’…|‚• (an ) (bn ) ‡~† sn = a1 + · · · + aP
n ‘~ ‰‚Ž†‡y ~…ŽŒ|
+∞
•‰~‘~ ‘„• •Ž›‘„•
P+∞ _Š „ (bn ) ‚|Š~† “…|ŠŒ’•~ ‡~† ‡y‘– “Ž~€‰zŠ„ ‡~† ~Š „ •‚†Žy n=1 an •’€‡ˆ|Š‚†
‘™‘‚ „ •‚†Žy n=1 an bn •’€‡ˆ|Š‚†
_•™•‚†‹„ e (bn ) •’€‡ˆ|Š‚† Œ•™‘‚ z•‘– bnP
→b
`y•‚† ‘Œ’ ‡Ž†‘„Ž|Œ’ ‘Œ’ 'LULFKOHW „ •‚†Žy +∞
n=1 an (bn − b) •’€‡ˆ|Š‚†
q›Ž~ †•”š‚† an bn = an (bn − b) + an b €†~ ‡y…‚ n Œ•™‘‚
P+∞
P+∞
P+∞
P+∞
P+∞
n=1 an bn =
n=1 an (bn − b) +
n=1 an b =
n=1 an (bn − b) + b
n=1 an ,
P+∞
Œ•™‘‚ „ n=1 an bn •’€‡ˆ|Š‚†
hogqeogm ck_ii_ppmjckvk
nompejvk _Š „ ~‡ŒˆŒ’…|~ (bn ) ‚|Š~† “…|ŠŒ’•~ ‡~†
P
n−1 b = b − b + b − b + · · · •’€‡ˆ|Š‚†
bn → 0 ‘™‘‚ „ •‚†Žy +∞
(−1)
n
1
2
3
4
n=1
(
P
0, ~Š n yŽ‘†Œ•
_•™•‚†‹„ c|Š~† nk=1 (−1)k−1 =
VŽ~ •y•‚† ‘Œ’ ‡Ž†‘„Ž|Œ’ ‘Œ’ 'LULFKOHW
1, ~Š n •‚Ž†‘‘™•
P
n−1 b •’€‡ˆ|Š‚†
„ •‚†Žy +∞
n
n=1 (−1)
P
(−1)n−1
n~Žy•‚†€‰~
q’•†‡y •~Ž~•‚|€‰~‘~ ‚|Š~† Œ† •‚†Žz• +∞
™‘~Š 0 < p ≤ 1 m†
n=1
np
•‚†Žz• ~’‘z• •’€‡ˆ|ŠŒ’Š ’•Œ •’Š…{‡„
P
P
(−1)n−1
(−1)n−1
√
= 1 − 12 + 31 − 41 + · · · ‡~† +∞
=
m† ~•ˆŒš•‘‚Ž‚• ~•™ ~’‘z• ‚|Š~† Œ† +∞
n=1
n=1
n
n
1−
√1
2
+
√1
3
−
√1
4
+ · · · X”Œ’‰‚ {•„ ~•Œ•‚|‹‚† ™‘† „ •Ž›‘„ •‚†Žy •’€‡ˆ|Š‚†
_•‡{•‚†•
P+∞ n!
3
n=1 n
n=1 3n
(n!)2 P+∞ 4n (n!)2
n=1 (2n)!
n=1 (2n)!
c“~Ž‰™•‘‚ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ ™•Œ’ ‚|Š~† •’Š~‘™
P+∞ 3n n! P+∞ en (n+1)! P+∞ 2·5·8···(3n−1) P+∞
n=1 nn
n=1
nn
n=1 1·5·9···(4n−3)
P+∞
3n P+∞ n!
n=1 n!
n=1 nn
P+∞ 10n n!(3n)!
n=1
(4n)!
P+∞
P+∞ 3 P+∞ n+1 n P+∞ n−1 2n
c“~Ž‰™•‘‚ ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~• ™•Œ’ ‚|Š~† •’Š~‘™
( 2n−1 )
( n+1 )
n=1 n
P+∞ n=1
P+∞ n=1
P+∞
P+∞ 3 n P+∞ 2n P+∞ √
n
n2
n n2
n
n
n
n
√
√
n=1 ( n n+1)n
n=1 e ( n+1 )
n=1 ( n n+1)n
n=1 nn
n=1 n 2
n=1 ( n − 1)
c‹‚‘y•‘‚ ‘„ •š€‡ˆ†•„ ‘–Š •‚†Ž›Š 12 + 31 + 212 + 312 + 213 + 313 + 214 + 314 + · · · ‡~† 21 + 1 +
+ 212 + 215 + 214 + 217 + 216 + · · · ‚“~Ž‰™ƒŒŠ‘~• ‘~ ‡Ž†‘{Ž†~ ˆ™€Œ’ ‡~† Ž|ƒ~•
P
(−1)n P+∞ (−1)n
c‹‚‘y•‘‚ –• •ŽŒ• ‘„ •š€‡ˆ†•„ ‡~† ‘„Š ~•™ˆ’‘„ •š€‡ˆ†•„ ‘†• +∞
n=2 n(ORJ n)2
n=2 n ORJ n
P+∞ (−1)n(n−1)/2 P+∞ (−1)n(n−1)/2 P+∞
1
1 P+∞
n−1
n−1
ORJ(1 + n1 )
n=1
n=1
n=1 (−1)
n=1 (−1)
2n
n
n ORJ(1 + n )
P+∞
P
n−1
2
P
P
n
+∞ (−1)
+∞
+∞
n−1 ORJ n
n−1 VLQ 1
n−1 n(1 − FRV 1 )
n=1 (−1)
n=1
n=1 (−1)
n=1 (−1)
n
3n
n
n
P+∞ (−1)n−1 P+∞ (−1)n−1
c‹‚‘y•‘‚ ‘†• n=1 3n+(−1)n n
n=1 3n+6(−1)n n –• •ŽŒ• ‘„ •š€‡ˆ†•„ ‡~† ‘„Š ~•™ˆ’‘„
•š€‡ˆ†•„
P+∞
P
P+∞ (−1)n−1
n−1 ( √
n−1 e − (1 + 1 )n
n
√
_•Œ•‚|‹‘‚ ™‘† Œ† +∞
(−1)
n − 1)
n=1 (−1)
n=1
n=1 n n n
n
•’€‡ˆ|ŠŒ’Š ’•™ •’Š…{‡„
P
1
_•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 1+xn •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š |x| > 1
P
P+∞
P+∞
>~@ _Š +∞
n=1 |xn | < +∞ ~•Œ•‚|‹‘‚ ™‘† Œ†
n=1 xn FRV(nx)
n=1 xn VLQ(nx) •’€‡ˆ|
ŠŒ’Š ~•Œˆš‘–•
>•@ _Š 0 ≤ r < 1 ~•Œ•‚|‹‘‚ ™‘†
P
P
1−r 2
2r VLQ x
n
n
1 + 2 +∞
2 +∞
n=1 r FRV(nx) = 1−2r FRV x+r2 ,
n=1 r VLQ(nx) = 1−2r FRV x+r 2 .
P
>€@ _Š „ (xn ) ‚|Š~† “…|ŠŒ’•~ ‡~†
xn → 0 ~•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 xn FRV(nx) •’€‡ˆ|Š‚† ~Š x 6=
P+∞
m2π €†~ ‡y…‚ m ∈ Z ‡~† ™‘† „ n=1 xn VLQ(nx) •’€‡ˆ|Š‚† €†~ ‡y…‚ x
P+∞
P
VLQ(nx) P+∞ ORJ n
1
1 VLQ(nx)
•’€‡ˆ|ŠŒ’Š
_•Œ•‚|‹‘‚ ™‘† Œ† +∞
n=1 n VLQ(nx)
n=1 (1 + 2 + · · · + n ) n
n=1 n ORJ n
€†~ ‡y…‚ x
1
23
R n+1 1
√
1
≥ 1
_•Œ•‚|‹‘‚ ™‘† 21 + 14 + · · · + 2n
2x dx = ORJ n + 1 €†~ ‡y…‚ n h~‘™•†Š •y•‚†
√1
‘„• 1 + x ≤ ex ~•Œ•‚|‹‘‚ ™‘† 12 · 43 · 56 · · · 2n−1
2n ≤ n+1 €†~ ‡y…‚ n
P
n−1 1·3·5···(2n−1) •’€‡ˆ|Š‚†
_•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 (−1)
2·4·6···(2n)
Rn 1
√
1
_•Œ•‚|‹‘‚ ™‘† 1 + 31 + · · · + 2n−1
≤ 1 + 1 2x−1
dx = 1 + ORJ 2n − 1 €†~ ‡y…‚ n h~‘™•†Š
√1
•y•‚† ‘„• 1 + x ≤ ex ~•Œ•‚|‹‘‚ ™‘† 1·3·5···(2n−1)
2·4·6···(2n) ≥ e 2n−1 €†~ ‡y…‚ n
P
n−1 1·3·5···(2n−1) •‚Š •’€‡ˆ|Š‚† ~•Œˆš‘–•
_•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 (−1)
2·4·6···(2n)
2 P+∞
P+∞
n−1 1·3·5···(2n−1) 3 –• •ŽŒ•
q| z”‚‘‚ Š~ •‚|‘‚ €†~ ‘†• n=1 (−1)n−1 1·3·5···(2n−1)
n=1 (−1)
2·4·6···(2n)
2·4·6···(2n)
‘„ •š€‡ˆ†•„ ‡~† ‘„Š ~•™ˆ’‘„ •š€‡ˆ†•„
P+∞
P+∞ _Š „ n=1 an •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† „ ~‡ŒˆŒ’…|~ (bn ) ‚|Š~† “Ž~€‰zŠ„ ~•Œ•‚|‹‘‚ ™‘† „
n=1 an bn •’€‡ˆ|Š‚† ~•Œˆš‘–•
P
n−1 b ‡~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~
X•‘– (bn ) “…|ŠŒ’•~ ›•‘‚ bn → 0 X•‘– s = +∞
n
n=1 (−1)
n−1
n
sn = b1 − b2 + · · · + (−1)
bn _•Œ•‚|‹‘‚ ™‘† 0 ≤ (−1) (s − sn ) ≤ bn+1 €†~ ‡y…‚ n
P
P+∞ an
an
′
>~@ X•‘– ™‘† „ +∞
n=1 np •’€‡ˆ|Š‚† _•Œ•‚|‹‘‚ ™‘† „
n=1 np′ •’€‡ˆ|Š‚† ~Š p > p
‡~† •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š p′ > p + 1
P
an
’•™ •’Š…{‡„ _•Œ•‚|‹‘‚ ™‘† ’•yŽ”Œ’Š p1 p2 ›•‘‚ p1 ≤ p0 ≤
>•@ X•‘– ™‘† „ +∞
n=1 np0 •’€‡ˆ|Š‚†
P+∞ an
p2 ‡~† p2 − p1 ≤ 1 ‡~† ›•‘‚ „ n=1 np Š~ ~•Œ‡ˆ|Š‚† ~Š p < p1 Š~ •’€‡ˆ|Š‚† ’•™ •’Š…{‡„ ~Š
p1 p2
P
u1 +···+un •’€‡ˆ|Š‚† ~•Œˆš‘–•
X•‘– un → u > 0 ‡~† |x| < 1 _•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 x
P
X•‘– ™‘† „ ~‡ŒˆŒ’…|~ (bn ) ‚|Š~† ‰ŒŠ™‘ŒŠ„ ‡~† bn → l _Š sn = nk=1 (−1)k−1 bk €†~
‡y…‚ n ~•Œ•‚|‹‘‚ ™‘† OLP sn − OLP sn = |l|
< 1 _•Œ•‚|‹‘‚
X•‘– m ∈ N ‰‚ m ≥ 2 X•‘– ™‘† †•”š‚† xn 6= 0 €†~ ‡y…‚ n ‡~† OLP xn+m
x
n
P+∞
™‘† „ n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–•
P
P+∞
X•‘– ™‘† †•”š‚† yn > 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† „ +∞
n=1 xn •’€‡ˆ|Š‚† ‡~† „
n=1 yn
~•Œ‡ˆ|Š‚† _•Œ•‚|‹‘‚ ™‘† OLP xynn ≤ 0 ≤ OLP xynn
~‡ŒˆŒ’…|~ (bn ) ›•‘‚ Š~ †•”š‚† bn > 0 €†~ ‡y…‚ n ›•‘‚ bn → 0 ‡~† ›•‘‚ „
P+∞ `Ž‚|‘‚
n−1
bn Š~ ~•Œ‡ˆ|Š‚†
n=1 (−1)
P+∞
`Ž‚|‘‚ n=1 an „ Œ•Œ|~
P •’€‡ˆ|Š‚† ‡~† ~‡ŒˆŒ’…|~ (bn ) ›•‘‚ bn → 0 ›•‘‚ Š~ †•”š‚†
bn ≥ 0 €†~ ‡y…‚ n ‡~† ›•‘‚ „ +∞
n=1 an bn Š~ ~•Œ‡ˆ|Š‚†
P+∞ 2
P
`Ž‚|‘‚ +∞
n=1 xn Š~ ~•Œ‡ˆ|Š‚†
n=1 xn „ Œ•Œ|~ •’€‡ˆ|Š‚† ›•‘‚ „
X•‘– ™‘† „ ~‡ŒˆŒ’…|~ (an ) ‚|Š~† •‚Ž†Œ•†‡{ b„ˆ~•{ ’•yŽ”‚† p ›•‘‚ Š~ †•”š‚† an+p = an
€†~ ‡y…‚ n X•‘– ‚•|•„• ™‘† „ ~‡ŒˆŒ’…|~ (bn ) ‚|Š~† “…|ŠŒ’•~ ‡~† bn → 0
P
_Š a1 + · · · + ap = 0 ~•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 an bn •’€‡ˆ|Š‚†
P+∞
P
_Š a1 + · · · + ap 6= 0 ~•Œ•‚|‹‘‚ ™‘† „ n=1 an bn •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š „ +∞
n=1 bn •’€‡ˆ|Š‚†
p
q
p
q
p
q
_•Œ•‚|‹‘‚ ™‘† „ 1 − 2 + 3 − 4 + 5 − 6 + · · · •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š p = q
_•Œ•‚|‹‘‚ ™‘† „ p1 − 2q + 3r + p4 − 5q + 6r + · · · •’€‡ˆ|Š‚† ~Š ‡~† ‰™ŠŒ ~Š p + r = q
_Ž€™‘‚Ž~ •‘~ •~Ž~•‚|€‰~‘~
‡~†
…~ •Œš‰‚ –• ‚†•†‡{ •‚Ž|•‘–•„ ‘Œ’ €‚Š†‡Œš •†–Š’‰†‡Œš ‘š•Œ’
‘Œ’ 1HZWRQ ™‘† ‘Œ y…ŽŒ†•‰~ ‘„• •‚†Žy• ~’‘{• ‚|Š~† Œ ~Ž†…‰™• 1 − √12
P
an
j†~ •„‰~Š‘†‡{ y•‡„•„ n‚Ž†€Žy“‚‘~† ‰†~ •†y‘~‹„ ‘–Š •‚†Ž›Š +∞
n=1 np •‚ •”z•„ ‰‚ ‘„ •š€‡ˆ†•„ ‡~† ‘„Š
~•™ˆ’‘„ •š€‡ˆ†•{ ‘Œ’• ~ŠyˆŒ€~ ‰‚ ‘†• ‘†‰z• ‘„• •~Ž~‰z‘ŽŒ’ p m† •‚†Žz• ~’‘z• ŒŠŒ‰yƒŒŠ‘~† •‚†Žz• 'LULFKOHW ‡~†
c•|•„• ‘„Š y•‡„•„
‡~† ‘„ •”‚‘†‡{
‚‰“~Š|ƒŒŠ‘~† •‘„Š _Š~ˆ’‘†‡{ f‚–Ž|~ _Ž†…‰›Š b‚|‘‚ ‘„Š y•‡„•„
’•Œ•„‰‚|–•„ €†~ ‘Œ •„‰~Š‘†‡™‘‚ŽŒ •~Žy•‚†€‰~ ‘z‘Œ†~• •‚†Žy• ‘„Š ζ •’ŠyŽ‘„•„ ‘Œ’ 5LHPDQQ
XŠ~ •’‰•ˆ{Ž–‰~ ‘Œ’ ‡Ž†‘„Ž|Œ’ ‚Š~ˆˆ~••™‰‚Š–Š •ŽŒ•{‰–Š
e •‚Ž|•‘–•„ m = 1 ‚|Š~† ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’
X•‘– ~‡ŒˆŒ’…|~ (xn ) ‡~† yn = n(xn − xn+1 ) €†~ ‡y…‚ n
P
P+∞
_Š nxn → a ‡~† a 6= 0 ~•Œ•‚|‹‘‚ ™‘† Œ† +∞
n=1 xn ‡~†
n=1 yn ~•Œ‡ˆ|ŠŒ’Š
P+∞
P+∞
P
P+∞
_Š Œ† n=1 xn ‡~† n=1 yn •’€‡ˆ|ŠŒ’Š ~•Œ•‚|‹‘‚ ™‘† nxn → 0 ‡~† +∞
n=1 xn =
n=1 yn
P+∞ 1
P+∞
1
_•Œ•‚|‹‘‚ ™‘† n=1 n2 = 2 − n=1 n(n+1)2
P
_•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š ‡~† ‰™ŠŒ ~Š €†~ ‡y…‚ ǫ > 0 ’•yŽ”‚† n0
›•‘‚ Š~ †•”š‚† |xn1 + · · · + xnk | < ǫ €†~ ‡y…‚ k ‡~† €†~ ‡y…‚ •†~“ŒŽ‚‘†‡Œš• n1 , . . . , nk ‰‚
n1 , . . . , n k ≥ n0
1
p
+ 1q = 1
P+∞
P+∞
P
q
p
>~@ _Š +∞
n=1 an bn •’€‡ˆ|Š‚† ~•Œˆš
n=1 |bn | < +∞ ~•Œ•‚|‹‘‚ ™‘† „
n=1 |an | < +∞ ‡~†
‘–• ‡~† ™‘† †•”š‚† „ ~Š†•™‘„‘~ ‘Œ’ +¶OGHU €†~ •‚†Žz•
P+∞
P+∞
P+∞
q 1/q .
p 1/p
n=1 an bn ≤
n=1 |bn |
n=1 |an |
X•‘– p, q > 1 ›•‘‚
_•Œ•‚|‹‘‚ ™‘† †•”š‚† „ †•™‘„‘~ ~Š ‡~† ‰™ŠŒ ~Š ’•yŽ”Œ’Š s, t ≥ 0 ™”† ‡~† Œ† •šŒ |•Œ† ‰‚ 0 ›•‘‚ Š~
†•”š‚†
s|an |p = t|bn |q €†~ ‡y…‚
an bn ≥ 0 €†~ ‡y…‚ n ‚|‘‚ an bn ≤ 0 €†~ ‡y…‚ n
P+∞
P n ‡~† ‚|‘‚
2 < +∞ ‚•‚†•{ 1 + 1 = 1 „ •Œˆš •„‰~Š‘†‡{ ~Š†•™‘„‘~
_Š n=1 |an |2 < +∞ ‡~† +∞
|b
|
n=1 n
2
2
‘Œ’ &DXFK\
P+∞
P+∞
P+∞
2 1/2 ,
2 1/2
n=1 |bn |
n=1 an bn ≤
n=1 |an |
‚|Š~† ‚†•†‡{ •‚Ž|•‘–•„ ‘„• ~Š†•™‘„‘~• ‘Œ’ +¶OGHU
P
P+∞
p
p
>•@ _Š +∞
n=1 |an | < +∞ ‡~†
n=1 |bn | < +∞ ~•Œ•‚|‹‘‚ ‘„Š ~Š†•™‘„‘~ ‘Œ’ 0LQNRZVNL €†~
•‚†Žz•
P+∞
P+∞
P+∞
p 1/p ≤
p 1/p +
p 1/p .
n=1 |an + bn |
n=1 |an |
n=1 |bn |
_•Œ•‚|‹‘‚ ™‘† †•”š‚† „ †•™‘„‘~ ~Š ‡~† ‰™ŠŒ ~Š ’•yŽ”Œ’Š s, t ≥ 0 ™”† ‡~† Œ† •šŒ |•Œ† ‰‚ 0 ›•‘‚ Š~
†•”š‚† s|an | = t|bn | €†~ ‡y…‚ n ‡~† ‚|‘‚ an bn ≥ 0 €†~ ‡y…‚ n ‚|‘‚ an bn ≤ 0 €†~ ‡y…‚ n
P
X•‘– ™‘† †•”š‚† wn > 0 €†~ ‡y…‚ n ‡~†P +∞
n=1 wn = 1 _Š „ f : I → R ‚|Š~† ‡’Ž‘{ •‘Œ
+∞
•†y•‘„‰~ I ‡~† †•”š‚† an ∈ I €†~ ‡y…‚ n ‡~† „ n=1 an wn •’€‡ˆ|Š‚† ~•Œ•‚|‹‘‚ ‘„Š ~Š†•™‘„‘~
‘Œ’ -HQVHQ €†~ •‚†Žz•
P+∞
P+∞
f
n=1 f (an )wn .
n=1 an wn ≤
_Š „ f ‚|Š~† ‡Œ|ˆ„ •‘Œ I ~•Œ•‚|‹‘‚ ™‘† †•”š‚† „ ~Š‘|•‘ŽŒ“„ ~Š†•™‘„‘~
_Š ‚•†•ˆzŒŠ „ f ‚|Š~† €Š„•|–• ‡’Ž‘{ ‡Œ|ˆ„ •‘Œ I ~•Œ•‚|‹‘‚ ™‘† „ ~Š†•™‘„‘~ ~’‘{ †•”š‚† –•
†•™‘„‘~ ~Š ‡~† ‰™ŠŒ ~Š Œ† ~Ž†…‰Œ| a1 , a2 , a3 , . . . ‚|Š~† |•Œ† ‰‚‘~‹š ‘Œ’•
~•Œ•‚|‹‘‚ ™‘† ™‘~Š ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ •‚|”Š‚† •š
tŽ„•†‰Œ•Œ†›Š‘~• ‘„Š y•‡„•„
€‡ˆ†•„ { ~•™‡ˆ†•„ •‚†Žy• ‘Œ |•†Œ •’‰•~|Š‚† ‡~† ‰‚ ‘Œ ‡Ž†‘{Ž†Œ Ž|ƒ~•
P+∞
X•‘–
P+∞f : N → N €Š„•|–• ~š‹Œ’•~ •‘Œ N _•™ ‡y…‚ •‚†Žy n=1 xn •„‰†Œ’Ž€Œš‰‚ ‰†~
Šz~ •‚†Žy n=1 yn –• ‚‹{• …z‘Œ’‰‚ y1 = x1 + · · · + xf (1) ‡~† yn = xf (n−1)+1 + · · · + xf (n) €†~
‡y…‚ n ≥ 2
P
P+∞
_•Œ•‚|‹‘‚ ™‘† ~Š „ +∞
n=1 xn z”‚† y…ŽŒ†•‰~ ‘™‘‚ „
n=1 yn z”‚† ‘Œ |•†Œ y…ŽŒ†•‰~
v•
~Š‘†•~Žy•‚†€‰~ €†~ ‘Œ ~Š‘|•‘ŽŒ“Œ …‚–Ž{•‘‚ ‘„Š f : N → N ‰‚ ‘š•Œ f (k) = 2k ‡~† ‘„ •‚†Žy
P+∞
n−1
n=1 (−1)
X•‘– ™‘†P
’•yŽ”‚† M ›•‘‚ Š~ †•”š‚† f (k +1)−f
(k) ≤ M €†~ ‡y…‚ k ‡~† z•‘– xn → 0 _•Œ•‚|‹‘‚
P
+∞
™‘† ~Š „ n=1
yn z”‚† y…ŽŒ†•‰~ ‘™‘‚ „ +∞
x
n=1 n z”‚† ‘Œ |•†Œ y…ŽŒ†•‰~
′
mŽ|ƒŒ’‰‚ ‚•†•ˆzŒŠ y1 = |x1 | + · · · + |xf (1) | ‡~† yn′ = |xf (n−1)+1 | + · · · + |xf (n) | €†~ ‡y…‚ n ≥ 2
P
P+∞
_•Œ•‚|‹‘‚ ™‘† ~Š yn′ → 0 ‡~† „ +∞
n=1 yn z”‚† y…ŽŒ†•‰~ ‘™‘‚ „
n=1 xn z”‚† ‘Œ |•†Œ y…ŽŒ†•‰~
e •’Šz”‚†~ ‘„• y•‡„•„•
P
e y•‡„•„ ~’‘{ •‚Ž†€Žy“‚†
•‚†Žy• iz‰‚ ™‘† „ •‚†Žy +∞
n=1 yn
P+∞ ‘ŒŠ ‰„”~Š†•‰™ ‘„• Œ‰~•Œ•Œ|„•„• ‘–Š ™Ž–Š ‰†~• P
+∞
•ŽŒ‡š•‘‚† ~•™ ‘„ •‚†Žy n=1 xn ‰‚ ‚†•~€–€{ •~Ž‚Š…z•‚–Š ‡~† ™‘† „ •‚†Žy n=1 xn •ŽŒ‡š•‘‚† ~•™ ‘„ •‚†Žy
P
+∞
n=1 yn ‰‚ •†~€Ž~“{ •~Ž‚Š…z•‚–Š
P
>~@ _Š „ (bn ) ‚|Š~† ‰ŒŠ™‘ŒŠ„ ‡~† “Ž~€‰zŠ„ ~•Œ•‚|‹‘‚ ™‘† +∞
n=1 |bn − bn+1 | < +∞
>•@ X•‘– ~‡ŒˆŒ’…|‚• (an ) (bn ) ‡~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = a1 + · · · + an ‘„• •Ž›‘„•
P
P+∞
_Š +∞
n=1 |bn −bn+1 | < +∞ ~Š bn → 0 ‡~† ~Š „ (sn ) ‚|Š~† “Ž~€‰zŠ„ ~•Œ•‚|‹‘‚ ™‘† „
n=1 an bn
•’€‡ˆ|Š‚†
P
P+∞
P+∞
_Š +∞
n=1 |bn − bn+1 | < +∞ ‡~† ~Š „
n=1 an •’€‡ˆ|Š‚† ~•Œ•‚|‹‘‚ ™‘† „
n=1 an bn •’€‡ˆ|Š‚†
>€@ _•Œ•‚|‹‘‚
P+∞™‘† „ ~‡ŒˆŒ’…|~ (bn ) ‚|Š~† •†~“ŒŽy •šŒ ‰ŒŠ™‘ŒŠ–Š “Ž~€‰zŠ–Š ~‡ŒˆŒ’…†›Š ~Š ‡~†
‰™ŠŒ ~Š n=1 |bn − bn+1 | < +∞
≤
>~@ X•‘– ™‘† †•”š‚† yn > 0 ‡~† xn 6= 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† †•”š‚† ‘‚ˆ†‡y xxn+1
n
P+∞
P+∞
yn+1
n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–•
n=1 yn •’€‡ˆ|Š‚† ~•Œ•‚|‹‘‚ ™‘† „
yn _Š „
−1+ µ → 0
>•@ X•‘– ™‘† †•”š‚† xn 6= 0 €†~ ‡y…‚ n ‡~† z•‘– ™‘† ’•yŽ”‚† µ ›•‘‚ n xxn+1
n
n
P
_•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š µ > 1 ‡~† •‚Š •’€‡ˆ|Š‚†~•Œˆš‘–• ~Š µ < 1
−1+ 1 + λ
X•‘– µ = 1 ‡~† z•‘– ™‘† ’•yŽ”‚† λ ›•‘‚ n ORJ n xxn+1
n
n ORJ n → 0 _•Œ•‚|‹‘‚ ™‘†
n
P+∞
„ n=1 xn •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š λ > 1 ‡~† •‚Š •’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š λ < 1
P
n−1 nn •’€‡ˆ|Š‚† ’•™ •’Š…{‡„
_•Œ•‚|‹‘‚ ™‘† „ +∞
n=1 (−1)
en n!
X•‘– ™‘† †•”š‚† |xn,m | ≤ yn €†~ ‡y…‚ P
n, m ‡~† OLPm→+∞ xn,mP= xn €†~ ‡y…‚ n
P
+∞
+∞
_Š +∞
y
<
+∞
~•Œ•‚|‹‘‚
™‘†
€†~
‡y…‚
m
„
n=1 n
n=1 xn,m •’€‡ˆ|Š‚† „
n=1 xn •’€‡ˆ|Š‚†
‚•|•„• ‡~† ‘zˆŒ•
P
P+∞
P+∞
OLPm→+∞ +∞
n=1 xn,m =
n=1 xn =
n=1 OLPm→+∞ xn,m .
P+∞
P
_Š „ •‚†Žy +∞
n=1 nxn ~•Œ‡ˆ|Š‚†
n=1 xn ~•Œ‡ˆ|Š‚† ~•Œ•‚|‹‘‚ ™‘† ‡~† „
b†~•Œ”†‡{ y…ŽŒ†•„ •†•ˆ›Š •‚†Ž›Š
mogpjmp
X•‘– •’ŠyŽ‘„•„ x : N × N → R b„ˆ~•{ •‚ ‡y…‚ ƒ‚š€Œ• (m, n) “’•†‡›Š
~Š‘†•‘Œ†”|ƒ‚‘~† ‰z•– ‘„• x Œ ~Ž†…‰™• x(m, n) hy…‚ ‘z‘Œ†~ •’ŠyŽ‘„•„ x ”~Ž~‡‘„Ž|ƒ‚‘~† •†•ˆ{
~‡ŒˆŒ’…|~ ‡~† ™•–• ‰‚ ‘†• •’Š{…‚†• ~‡ŒˆŒ’…|‚• •ŽŒ‘†‰y‰‚ ‘Œ •š‰•ŒˆŒ xm,n ~Š‘| ‘Œ’ x(m, n)
b„ˆ~•{
xm,n = x(m, n).
a†~ ‘„ •†•ˆ{ ~‡ŒˆŒ’…|~ x ”Ž„•†‰Œ•Œ†Œš‰‚ ‡~† ‘~ •š‰•Œˆ~
(xm,n )
{
(xm,n )+∞
m,n=1 .
p ~’‘{Š ‘„Š ‚Š™‘„‘~ …~ ‰~• ~•~•”Œˆ{•‚† ‘Œ …z‰~ ‘„• y…ŽŒ†•„• •†•ˆ›Š ~‡ŒˆŒ’…†›Š qŒ ~Š‘|
•‘Œ†”Œ …z‰~ €†~ ‘†• •’Š{…‚†• ~‡ŒˆŒ’…|‚• ‚|Š~† ‘Œ …z‰~ ~’‘Œš ‘Œ’ ‡‚“~ˆ~|Œ’ •„ˆ~•{ Œ† •‚†Žz•
VŽ~ • ~’‘{Š ‘„Š ‚Š™‘„‘~ …~ ~•”Œˆ„…Œš‰‚ ‰‚ ‘†• ˆ‚€™‰‚Š‚• •†•ˆz• •‚†Žz• h~† ‰yˆ†•‘~ •‚Š …~
‚‹‚‘y•Œ’‰‚ ‘„ €‚Š†‡{ …‚–Ž|~ ‘–Š •†•ˆ›Š •‚†Ž›Š ‡~† ‘Œ’• •ŒˆˆŒš• •†~“ŒŽ‚‘†‡Œš• ‘Ž™•Œ’• y…ŽŒ†
•{• ‘Œ’• f~ ‚‹‚‘y•Œ’‰‚ ‰™ŠŒ zŠ~ ‚†•†‡™ ~ˆˆy ~Ž‡‚‘y ”Ž{•†‰Œ …z‰~ ‘Œ …z‰~ ‘„• ˆ‚€™‰‚Š„•
•†~•Œ”†‡{• y…ŽŒ†•„• •†•ˆ›Š •‚†Ž›Š ‡~† …~ •Œš‰‚ ‰™ŠŒ •šŒ •~•†‡y ~•Œ‘‚ˆz•‰~‘~
mogpjmp
q~ •š‰•Œˆ~ •Œ’ ”Ž„•†‰Œ•Œ†Œš‰‚ €†~ ‰†~ •†•ˆ{ •‚†Žy ‚|Š~†
P
P+∞
(m,n)∈N×N xm,n .
m,n=1 xm,n {
P
iz‰‚ ™‘† „ ~‡ŒˆŒ’…|~ (bn ) z”‚† “Ž~€‰zŠ„ ‡š‰~Š•„ ~Š +∞
n=1 |bn −bn+1 | < +∞ p‘Œ >•@ •‚Ž†€Žy“ŒŠ‘~† •Ž›‘~ ‘Œ
‡Ž†‘{Ž†Œ ‘Œ’ 'HGHNLQG ‡~† ‡~‘™•†Š ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ 'X%RLV 5H\PRQG n~Ž~‘„Ž{•‘‚ ™‘† ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ 'HGHNLQG
‚|Š~† €‚Š|‡‚’•„ ‘Œ’ ‡Ž†‘„Ž|Œ’ ‘Œ’ 'LULFKOHW ‡~† ‘Œ ‡Ž†‘{Ž†Œ ‘Œ’ 'X%RLV 5H\PRQG ‚|Š~† €‚Š|‡‚’•„ ‘Œ’ ‡Ž†‘„Ž|Œ’ ‘Œ’
$EHO
e •’Šz”‚†~ ‘„• y•‡„•„•
qŒ ‡Ž†‘{Ž†Œ ‘Œ’ *DXVV XŠ~ ”Ž{•†‰Œ ‡Ž†‘{Ž†Œ •‚ •‚Ž†•‘›•‚†• •Œ’ ‘Œ ‡Ž†‘{Ž†Œ ˆ™€Œ’ ‘Œ’ G $OHPEHUW ~•Œ‘’€”yŠ‚†
e •†Œ ”Ž{•†‰„ ‰z…Œ•Œ• ‚Š~ˆˆ~€{• ‘–Š •’‰•™ˆ–Š ‘„• •‚†Žy• ‡~† ‘Œ’ ŒŽ|Œ’ ~‡ŒˆŒ’…|~•
mogpjmp
m •Ž›‘Œ• ‘Ž™•Œ• •†~•Œ”†‡{• y…ŽŒ†•„• •†•ˆ{• •‚†Žy• ‚|Š~† „ y…ŽŒ†•„ •Ž›‘~
‡~‘y €Ž~‰‰z•
_’‘™ •„‰~|Š‚† ™‘† •Ž›‘~ •Ž|•‡Œ’‰‚ €†~ ‡y…‚ m ∈ P
N ~Š ’•yŽ”‚† ‘Œ y…ŽŒ†•‰~
P
+∞
sm = +∞
x
‡~†
‡~‘™•†Š
•Ž|•‡Œ’‰‚
~Š
’•yŽ”‚†
‘Œ
y…ŽŒ†•‰~
m,n
n=1
m=1 sm •„ˆ~•{ ‘Œ
P+∞ P+∞
m=1
n=1 xm,n .
_Š ‘Œ ‘‚ˆ†‡™ ~•Œ‘zˆ‚•‰~ ‚|Š~† zŠ~• ~Ž†…‰™• a { ‘Œ +∞ { ‘Œ −∞ ‘™‘‚ ˆz‰‚ ™‘† „ •†•ˆ{ •‚†Žy
•’€‡ˆ|Š‚† •‘ŒŠ a { ~•Œ‡ˆ|Š‚† •‘Œ +∞ { ~•Œ‡ˆ|Š‚† •‘Œ −∞ ~Š‘†•‘Œ|”–• ‰‚ y…ŽŒ†•„ •Ž›‘~ ‡~‘y
€Ž~‰‰z•
m •‚š‘‚ŽŒ• ‘Ž™•Œ• •†~•Œ”†‡{• y…ŽŒ†•„• •†•ˆ{• •‚†Žy• ‚|Š~† „ y…ŽŒ†•„ •Ž›‘~ ‡~‘y
P •‘{ˆ‚• _’‘™
•„‰~|Š‚† ™‘† •Ž›‘~ •Ž|•‡Œ’‰‚ €†~ ‡y…‚ n ∈ P
N ~Š ’•yŽ”‚† ‘Œ y…ŽŒ†•‰~ tn = +∞
m=1 xm,n ‡~†
+∞
‡~‘™•†Š •Ž|•‡Œ’‰‚ ~Š ’•yŽ”‚† ‘Œ y…ŽŒ†•‰~ n=1 tn •„ˆ~•{ ‘Œ
P+∞ P+∞
n=1
m=1 xm,n .
_Š ‘Œ ‘‚ˆ†‡™ ~•Œ‘zˆ‚•‰~ ‚|Š~† zŠ~• ~Ž†…‰™• a { ‘Œ +∞ { ‘Œ −∞ ‘™‘‚ ˆz‰‚ ™‘† „ •†•ˆ{ •‚†Žy
•’€‡ˆ|Š‚† •‘ŒŠ a { ~•Œ‡ˆ|Š‚† •‘Œ +∞ { ~•Œ‡ˆ|Š‚† •‘Œ −∞ ~Š‘†•‘Œ|”–• ‰‚ y…ŽŒ†•„ •Ž›‘~ ‡~‘y
•‘{ˆ‚•
m ‘Ž|‘Œ• ‘Ž™•Œ• •†~•Œ”†‡{• y…ŽŒ†•„• •†•ˆ{• •‚†Žy• ‚|Š~† „ y…ŽŒ†•„P
•Ž›‘~ ‡~‘y •†~€–Š|Œ’•
k
_’‘™ •„‰~|Š‚† ™‘† •Ž›‘~ •Ž|•‡Œ’‰‚ €†~
‡y…‚
k
∈
N
‘Œ
y…ŽŒ†•‰~
u
=
k
l=1 xk−l+1,l ‡~† ‡~‘™•†Š
P+∞
•Ž|•‡Œ’‰‚ ~Š ’•yŽ”‚† ‘Œ y…ŽŒ†•‰~ k=1 uk •„ˆ~•{ ‘Œ
P+∞ Pk
k=1
l=1 xk−l+1,l
.
_Š ‘Œ ‘‚ˆ†‡™ ~•Œ‘zˆ‚•‰~ ‚|Š~† zŠ~• ~Ž†…‰™• a { ‘Œ +∞ { ‘Œ −∞ ‘™‘‚ ˆz‰‚ ™‘† „ •†•ˆ{ •‚†Žy
•’€‡ˆ|Š‚† •‘ŒŠ a { ~•Œ‡ˆ|Š‚† •‘Œ +∞ { ~•Œ‡ˆ|Š‚† •‘Œ −∞ ~Š‘†•‘Œ|”–• ‰‚ y…ŽŒ†•„ •Ž›‘~ ‡~‘y
•†~€–Š|Œ’•
m† ™ŽŒ† •‘{ˆ‚•
€Ž~‰‰z• ‡~† •†~€›Š†Œ† •ŽŒzŽ”ŒŠ‘~† •ŽŒ“~Š›• ~•™ ‘„ …‚–Ž|~ ‘–Š
•†Šy‡–Š s~Š‘~ƒ™‰~•‘‚ ™‘† Œ† ~Ž†…‰Œ| x1,1 , x1,2 , x1,3 , . . . •†~‘y••ŒŠ‘~† •‘„Š •Ž›‘„ •‚†Žy ~•™
‘~ ~Ž†•‘‚Žy •ŽŒ• ‘~ •‚‹†y _‰z•–• ~•™ ‡y‘– •†~‘y••ŒŠ‘~† Œ† ~Ž†…‰Œ| x2,1 , x2,2 , x2,3 , . . . ~‰z•–•
~•™ ‡y‘– ~•™ ~’‘Œš• •†~‘y••ŒŠ‘~† Œ† x3,1 , x3,2 , x3,3 , . . . ‡~† Œš‘– ‡~… ‚‹{• X‘•† •„‰†Œ’Ž€‚|‘~†
zŠ~• y•‚†ŽŒ• •|Š~‡~• ‰‚ ‘Œ •‘Œ†”‚|Œ x1,1 •‘„Š •yŠ– ~Ž†•‘‚Ž{ €–Š|~ ‘Œ’ ‡~† Œ Œ•Œ|Œ• ‚‡‘‚|Š‚‘~†
~•‚Ž†™Ž†•‘~ •ŽŒ• ‘~ •‚‹†y ‡~† •ŽŒ• ‘~ ‡y‘– qŒ •‘Œ†”‚|Œ xm,n ‚|Š~† •‘„Š ‘Œ‰{ ‘„• m Œ•‘{•
€Ž~‰‰{• ‡~† ‘„• n Œ•‘{• •‘{ˆ„•
[‘~Š z”Œ’‰‚ •‚•‚Ž~•‰zŠ~ ~…ŽŒ|•‰~‘~ ‘~ •Žy€‰~‘~ ‚|Š~† ~•ˆy a†~ •~Žy•‚†€‰~ †•”š‚†
PN
PM
PN
PM
m=1 xm,n
n=1 xm,n =
n=1
m=1
•†™‘† „ •‚†Žy ‰‚ ‘„Š Œ•Œ|~ €|Š‚‘~† „ •Ž™•…‚•„ •‚•‚Ž~•‰zŠŒ’ •ˆ{…Œ’• ~Ž†…‰›Š •‚Š ‚•„Ž‚yƒ‚† ‘„Š
‘†‰{ ‘Œ’ ~…ŽŒ|•‰~‘Œ• b‚|‘‚ ™‰–• ‘Œ ‚‹{• •~Žy•‚†€‰~


~Š m − n = 1
1,
n~Žy•‚†€‰~
X•‘– xm,n = −1, ~Š m − n = −1


0,
~Š m − n 6= ±1
c•› Œ y•‚†ŽŒ• •|Š~‡~• z”‚† •‘Œ†”‚|~ 1 •‘„Š •†~€›Š†Œ ~‡Ž†•›• ‡y‘– ~•™ ‘„Š ‡šŽ†~ •†~€›Š†Œ ‡~†
•‘Œ†”‚|~
−1 •‘„Š •†~€›Š†Œ ~‡Ž†•›•
‚|Š~†
0
P+∞~ˆˆŒ
P•‘Œ†”‚|Œ
P+∞ •yŠ– ~•™ ‘„Š ‡šŽ†~ •†~€›Š†Œ hy…‚
P+∞
+∞
x
x
=
0
€†~
‡y…‚
m
≥
2
Œ•™‘‚
x
=
−1
‡~†
q™‘‚
m=1
n=1 m,n = −1
n=1 m,n
n=1 1,n
P
P
P
+∞
+∞
+∞ P+∞
c•|•„•
xm,1 = 1 ‡~† m=1 xm,n = 0 €†~ ‡y…‚ n ≥ 2 Œ•™‘‚ n=1
m=1
m=1 xm,n = 1
P+∞ Pk
Pk
qzˆŒ• ‚|Š~† l=1 xk−l+1,l = 0 €†~ ‡y…‚ k Œ•™‘‚ k=1
l=1 xk−l+1,l = 0
c•Œ‰zŠ–• Œ† ‘Ž‚†• ‘Ž™•Œ† y…ŽŒ†•„• ‘„• •†•ˆ{• •‚†Žy• •|ŠŒ’Š ‘Ž|~ •†~“ŒŽ‚‘†‡y ~•Œ‘‚ˆz•‰~‘~
X•‘– •†•ˆ{ •‚†Žy „ Œ•Œ|~ z”‚† ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• •„ˆ~•{ z•‘– ™‘† †•”š‚† xm,n ≥ 0 €†~
‡y…‚ m, n
P
q™‘‚ €†~ ‡y…‚ m ‘Œ y…ŽŒ†•‰~ sm = +∞
n=1 xm,n ’•yŽ”‚† ‡~† ‚|Š~† ‚|‘‚ ~Ž†…‰™• ≥ 0 ‚|‘‚ +∞
q›Ž~ ‰†~ P
•Ž›‘„ •‚Ž|•‘–•„ ‚|Š~† ™‘~Š Œ sm ‚|Š~† ~Ž†…‰™• €†~ ‡y…‚ m q™‘‚ ™•–• €Š–Ž|ƒŒ’‰‚ ‘Œ
y…ŽŒ†•‰~ +∞
m=1 sm ’•yŽ”‚† ‡~† ‚|Š~† ‚|‘‚ ~Ž†…‰™• ≥ 0 ‚|‘‚ +∞ j†~ •‚š‘‚Ž„ •‚Ž|•‘–•„ ‚|Š~†
™‘~Š €†~ ‘Œ’ˆy”†•‘ŒŠ P
zŠ~Š m0 ‚|Š~† sm0 = +∞ n~Ž~‘„Ž{•‘‚ ‘™‘‚ ™‘† ™‘~Š ’•ŒˆŒ€|ƒŒ’‰‚ zŠ~
‰‚Ž†‡™ y…ŽŒ†•‰~ ‘„• +∞
m=1 sm ~’‘™ ‚|Š~† zŠ~ •’Š„…†•‰zŠŒ y…ŽŒ†•‰~ •‚•‚Ž~•‰zŠŒ’ •ˆ{…Œ’•
•‘Œ†”‚|–Š •‘Œ Œ•Œ|Œ •‚Š •ŽŒ‡š•‘‚† ~•ŽŒ••†™Ž†•‘„ ‰ŒŽ“{ ~“Œš ™ˆ~ ~’‘y ‘~ •‘Œ†”‚|~ ~Š{‡Œ’Š
•‘Œ [0, +∞] h~† ‚†•†‡›‘‚Ž~ €†~ ‡y…‚ k ≥ m0 ‚|Š~† s1 + · · · + sk = +∞ ~“Œš ‡y…‚ ‘z‘Œ†Œ
y…ŽŒ†•‰~ P
•‚Ž†z”‚† ‘ŒŠ ™ŽŒ sm0 c•Œ‰zŠ–• ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ z”Œ’Š ™Ž†Œ ‘Œ +∞ Œ•™‘‚ ‘Œ
+∞
y…ŽŒ†•‰~
+∞ p’‰•‚Ž~|ŠŒ’‰‚ ™‘† •‚ ‡y…‚ •‚Ž|•‘–•„ ‘Œ y…ŽŒ†•‰~
m=1 sm ’•yŽ”‚†
P+∞
P+∞ ‡~†
P‚|Š~†
+∞
‡~† ‚|Š~† ‚|‘‚ ~Ž†…‰™• ≥ 0 ‚|‘‚ +∞
m=1 sm •„ˆ~•{ ‘Œ
m=1
n=1 xm,n ’•yŽ”‚†
P+∞
P
_‡Ž†•›• ‘~ |•†~ †•”šŒ’Š ‡~† €†~ ‘Œ y…ŽŒ†•‰~ +∞
m=1 xm,n •„ˆ~•{ ™‘† ‚|Š~† ‚|‘‚ ~Ž†…‰™•
n=1
≥ 0 ‚|‘‚ +∞
e ‡~‘y•‘~•„ ‰‚ ‘„Š y…ŽŒ†•„ •Ž›‘~ ‡~‘y •†~€–Š|Œ’• ‚|Š~† ˆ|€Œ •†Œ ~•ˆ{
k ‘Œ y…ŽŒ†•‰~
P+∞a†~P‡y…‚
P
k
k
x
‚|Š~†
•ŽŒ“~Š›•
~Ž†…‰™•
≥
0
Œ•™‘‚
‘Œ
y…ŽŒ†•‰~
x
l=1 k−l+1,l
k=1
l=1 k−l+1,l ‚|Š~†
‚|‘‚ ~Ž†…‰™• ≥ 0 ‚|‘‚ +∞
m•™‘‚ •ŽŒ‡š•‘‚† ‘Œ ‚Ž›‘„‰~ ~Š ‘~ ‘Ž|~ ~…ŽŒ|•‰~‘~ •”‚‘|ƒŒŠ‘~† ‡~† ‚†•†‡›‘‚Ž~ ~Š ‚|Š~† |•~
p‘Œ ‚Ž›‘„‰~ ~’‘™ ~•~Š‘y ‘Œ …‚›Ž„‰~
fcvoej_
X•‘– ™‘† †•”š‚† xm,n ≥ 0 €†~ ‡y…‚ m, n q™‘‚
P+∞ P+∞
P+∞ Pk
P+∞ P+∞
m=1
n=1 xm,n =
n=1
m=1 xm,n =
k=1
l=1 xk−l+1,l .
b„ˆ~•{ €†~ •‚†Žz• ‰‚ ‰„ ~ŽŠ„‘†‡Œš• ™ŽŒ’• Œ† ‘Ž‚†• ‘Ž™•Œ† y…ŽŒ†•„• •|ŠŒ’Š ‘Œ |•†Œ ~•Œ‘zˆ‚•‰~ ‘Œ
Œ•Œ|Œ ‚|Š~† ‚|‘‚ ~Ž†…‰™• ≥ 0 ‚|‘‚ +∞
_•™•‚†‹„ a†~ ‡y…‚ m ‚|Š~†
P
P+∞
P+∞
s1 + · · · + sm = +∞
n=1 x1,n + · · · +
n=1 xm,n =
n=1 (x1,n + · · · + xm,n )
P+∞ P+∞
≤ n=1
m=1 xm,n .
P+∞ P+∞
b„ˆ~•{ ‘Œ y…ŽŒ†•‰~ n=1
m=1 xm,n ‚|Š~† yŠ– “Žy€‰~ ‘–Š ‰‚Ž†‡›Š ~…ŽŒ†•‰y‘–Š s1 +
· · · + sm €†~ ‡y…‚ m Œ•™‘‚
P+∞
P+∞ P+∞
P+∞ P+∞
m=1
n=1 xm,n =
m=1 sm ≤
n=1
m=1 xm,n .
j‚ ‘ŒŠ |•†Œ ‘Ž™•Œ •Ž|•‡Œ’‰‚ ‘„ •’‰‰‚‘Ž†‡{ •”z•„
P+∞ P+∞
P+∞ P+∞
n=1 xm,n
m=1 xm,n ≤
m=1
n=1
P+∞ P+∞
P+∞
P
‡~† ‡~‘~ˆ{€Œ’‰‚ •‘„Š †•™‘„‘~ +∞
m=1 xm,n
n=1 xm,n =
n=1
m=1
q›Ž~ •„‰†Œ’Ž€Œš‰‚ ‰†~ Šz~ •†•ˆ{ •‚†Žy ŒŽ|ƒŒŠ‘~•
(
xm,n−m+1 , ~Š m ≤ n
ym,n =
0,
~Š m > n
c•‚†•{ †•”š‚†Pym,n ≥ 0 €†~ ‡y…‚ m, n ‰•ŒŽŒš‰‚ Š~ ‚“~Ž‰™•Œ’‰‚ ‘Œ ‰z”Ž† ‘›Ž~ ~•Œ‘zˆ‚•‰~ •‘„
•†•ˆ{ •‚†Žy +∞
m,n=1 ym,n b„ˆ~•{
P+∞ P+∞
P+∞ P+∞
m=1
n=1 ym,n =
n=1
m=1 ym,n .
n~Ž~‘„ŽŒš‰‚ ™‰–• ™‘† ‘~ •šŒ ‰zˆ„ ‘„•
‚|Š~†
P+∞ P+∞
P+∞ P+∞
m=1
n=1 ym,n =
m=1
n=1 xm,n
‡~†
P+∞ P+∞
P
Pk
= +∞
k=1
l=1 xk−l+1,l .
P+∞ Pk
P+∞
n=1 xm,n =
k=1
l=1 xk−l+1,l
n=1
VŽ~ „
€|Š‚‘~†
P+∞
m=1
m=1 ym,n
mogpjmp
P+∞
m=1
iz‰‚ ™‘† „ •†•ˆ{ •‚†Žy
P+∞
n=1 |xm,n |
=
P+∞
m,n=1 xm,n
P+∞ P+∞
n=1
m=1 |xm,n |
•’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š
=
P+∞ Pk
l=1 |xk−l+1,l |
k=1
< +∞.
P
nŽz•‚† Š~ ‘ŒŠ|•Œ’‰‚ ™‘† ™‘~Š …zˆŒ’‰‚ Š~ •Œš‰‚ ~Š ‰†~ •†•ˆ{ •‚†Žy +∞
m,n=1 xm,n •’€‡ˆ|
P+∞
P+∞
Š‚† ~•Œˆš‘–• ‘™‘‚ ~Ž‡‚| Š~ •Œš‰‚ ~Š zŠ~ ‰™ŠŒ ~•™ ‘~ ‘Ž|~ ~…ŽŒ|•‰~‘~ m=1
|xm,n |
n=1
P+∞ P+∞
P+∞ Pk
n=1
m=1 |xm,n | ‡~†
k=1
l=1 |xk−l+1,l | ‚|Š~† ~Ž†…‰™• ‡~† ™”† +∞ •†™‘† ‘~ ‘Ž|~
~…ŽŒ|•‰~‘~ ‚|Š~† •š‰“–Š~ ‰‚ ‘Œ …‚›Ž„‰~
|•~
P+∞
fcvoej_
_Š „ •†•ˆ{ •‚†Žy m,n=1 xm,n •’€‡ˆ|Š‚† ~•Œˆš‘–• ‘™‘‚
P+∞
m=1
P+∞
n=1 xm,n
=
P+∞ P+∞
m=1 xm,n
n=1
=
P+∞ Pk
k=1
l=1 xk−l+1,l
‡~† „ ‡Œ†Š{ ‘†‰{ ‘–Š ‘Ž†›Š ~…ŽŒ†•‰y‘–Š ‚|Š~† ~Ž†…‰™•
P
P+∞
_•™•‚†‹„ _•™ ‘Œ +∞
|xm,n | < +∞ •’Š‚•y€‚‘~† •ŽŒ“~Š›• ™‘† €†~ ‡y…‚ m †•”š‚†
m=1
n=1
P+∞
P+∞
P+∞
n=1 |xm,n | < +∞ Œ•™‘‚ „ Pn=1 xm,n •’€‡ˆ|Š‚† ‡~† ‘Œ y…ŽŒ†•‰~ sm = P n=1 xm,n ‚|Š~†
+∞
~Ž†…‰™• m‰Œ|–• €†~ ‡y…‚ n „ +∞
m=1 xm,n •’€‡ˆ|Š‚† ‡~† ‘Œ y…ŽŒ†•‰~ tn =
m=1 xm,n ‚|Š~†
~Ž†…‰™•
P
P
P+∞
h~‘™•†Š €†~ ‡y…‚ m †•”š‚† |sm | = +∞
xm,n ≤ +∞
|xm,n | ‡~† ‚•Œ‰zŠ–•
n=1
n=1
m=1 |sm | ≤
P+∞ P+∞
P+∞
m=1
m=1 sm •’€‡ˆ|Š‚† ‡~† ‘Œ
n=1 |xm,n | < +∞ VŽ~ „
P
P+∞ P+∞
s = +∞
m=1 sm =
m=1
n=1 xm,n
‚|Š~† ~Ž†…‰™•
P
m‰Œ|–• €†~ ‡y…‚ n „ +∞
n=1 tn •’€‡ˆ|Š‚† ‡~† ‘Œ
P
P+∞ P+∞
t = +∞
n=1 tn =
n=1
m=1 xm,n
‚|Š~† ~Ž†…‰™•
q›Ž~ •Žz•‚† Š~ ~•Œ•‚|‹Œ’‰‚
s=t
P+∞ P+∞
P+∞
P+∞ ™‘†
X•‘– ǫ > 0 c•‚†•{ m=1
m=1 |xm,n | < +∞ •’Š‚•y
n=1
n=1 |xm,n | < +∞ ‡~†
€‚‘~† ™‘† ’•yŽ”Œ’Š m0 , n0 ›•‘‚ ~Š‘†•‘Œ|”–•
ǫ
ǫ
P+∞
P+∞
P+∞
P+∞
n=n0 +1
m=1 |xm,n | < 4 .
m=m0 +1
n=1 |xm,n | < 4 ,
q›Ž~ €Žy“Œ’‰‚
P+∞ P+∞
Pm0 Pn0
P 0 P n0
s− m
m=1
n=1 xm,n =
m=1
n=1 xm,n −
m=1
n=1 xm,n
P+∞
P+∞
P 0 P+∞
= m
n=1 xm,n
m=m0 +1
n=1 xm,n +
m=1
P 0 Pn 0
− m
n=1 xm,n
m=1
P+∞
P 0 P+∞
Pn0
P+∞
= m
m=1
n=1 xm,n −
n=1 xm,n +
m=m0 +1
n=1 xm,n
P+∞
P+∞
P 0 P+∞
= m
n=1 xm,n
n=n0 +1 xm,n +
m=m0 +1
m=1
P+∞
P
Pm 0
P+∞
= +∞
n=n0 +1
m=1 xm,n +
m=m0 +1
n=1 xm,n ,
Œ•™‘‚ ~•™ ‘†•
•’Š‚•y€‚‘~†
P+∞
P m0
P+∞
s − Pm0 Pn0 xm,n ≤ P+∞
m=1
n=1
n=n0 +1
m=1 |xm,n | +
m=m0 +1
n=1 |xm,n |
P+∞
P+∞
P+∞
P
≤ +∞
n=1 |xm,n |
m=1 |xm,n | +
m=m0 +1
n=n0 +1
<
ǫ
4
+
ǫ
4
=
ǫ
2
.
j‚ ‘ŒŠ |•†Œ ‘Ž™•Œ ‰•ŒŽ‚| Š~ ~•Œ•‚†”‘‚| ‡~† „ ~ŠyˆŒ€„ •”z•„
ǫ
P
Pm 0
t − n0
< .
n=1
m=1 xm,n
2
q›Ž~ •~Ž~‘„ŽŒš‰‚ ™‘†
P m0
m=1
_•™ ‘†•
|s − t| ≤ s −
‡~†
Pm 0
m=1
P n0
n=1 xm,n
=
P n0
n=1
P m0
m=1 xm,n
.
z”Œ’‰‚
Pn0 Pm0
Pn 0
<
n=1
n=1 xm,n + t −
m=1 xm,n
ǫ
2
+
ǫ
2
= ǫ.
_•Œ•‚|‹~‰‚ ™‘† †•”š‚† |s − t| < ǫ €†~ ‡y…‚ ǫ > 0 Œ•™‘‚ s = t
(
xm,n−m+1 , ~Š m ≤ n
q›Ž~ ™•–• •‘„Š ~•™•‚†‹„ ‘Œ’ …‚–Ž{‰~‘Œ•
ŒŽ|ƒŒ’‰‚ ym,n =
0,
~Š m > n
P+∞ P+∞
X‘•† •ŽŒ‡š•‘‚† ‰†~ Šz~ •†•ˆ{ •‚†Žy „ Œ•Œ|~ •’€‡ˆ|Š‚† ~•Œˆš‘–• •†™‘† m=1
|ym,n | =
n=1
P+∞ P+∞
|xm,n | < +∞ VŽ~ ‰•ŒŽŒš‰‚ Š~ ‚“~Ž‰™•Œ’‰‚ ‘Œ ‰z”Ž† ‘›Ž~ ~•Œ‘zˆ‚•‰~ •‘„
m=1
n=1
P
•†•ˆ{ •‚†Žy +∞
m,n=1 ym,n b„ˆ~•{
P+∞
m=1
P+∞
n=1 ym,n
[•–• •‘„Š ~•™•‚†‹„ ‘Œ’ …‚–Ž{‰~‘Œ•
P+∞ P+∞
‡~†
P+∞ P+∞
m=1 ym,n
n=1
VŽ~
m=1
P+∞
n=1 xm,n
=
=
P+∞ P+∞
.
l=1 xk−l+1,l
m=1 ym,n
n=1
‚|Š~†
P+∞ P+∞
m=1
n=1 xm,n
n=1 ym,n =
m=1
P+∞
P+∞ Pk
k=1
=
P+∞ Pk
k=1
l=1 xk−l+1,l
.
_•‡{•‚†•
P
X•‘– ™‘† †•”š‚† 0 ≤ xm,n ≤ ym,n €†~ ‡y…‚ m, n _Š „ •†•ˆ{ •‚†Žy +∞
m,n=1 ym,n •’
€‡ˆ|Š‚† –• •ŽŒ• Œ•Œ†ŒŠ•{•Œ‘‚ ~•™ ‡~† ‚•Œ‰zŠ–•P–• •ŽŒ• ™ˆŒ’• ‘Œ’• ‘Ž‚†• ‘Ž™•Œ’• y…ŽŒ†•„•
~•Œ•‚|‹‘‚ ™‘† ‘Œ |•†Œ †•”š‚† ‡~† €†~ ‘„ •†•ˆ{ •‚†Žy +∞
m,n=1 xm,n _Š sx ‚|Š~† „ ‡Œ†Š{ ‘†‰{ ‘–Š •†~
P+∞
•Œ”†‡›Š ~…ŽŒ†•‰y‘–Š ‘„• m,n=1 xm,n ‡~† sy ‚|Š~† „ ‡Œ†Š{ ‘†‰{ ‘–Š •†~•Œ”†‡›Š ~…ŽŒ†•‰y‘–Š
P
‘„• +∞
m,n=1 ym,n ~•Œ•‚|‹‘‚ ™‘† sx ≤ sy
`Ž‚|‘‚ ‘„Š ‘†‰{ ‘–Š ‘Ž†›Š •†~•Œ”†‡›Š ~…ŽŒ†•‰y‘–Š ‘„• •†•ˆ{• •‚†Žy•
_•Œ•‚|‹‘‚ ™‘† „ •†•ˆ{ •‚†Žy
_•Œ•‚|‹‘‚ ™‘† „ •†•ˆ{ •‚†Žy
P+∞
1
m,n=1 mp nq
P+∞
1
m,n=1 (m+n)!
•’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š ‡~† ‰™ŠŒ ~Š p, q > 1
P+∞
1
m,n=1 (m+n)p
•’€‡ˆ|Š‚† ~•Œˆš‘–• ~Š ‡~† ‰™ŠŒ ~Š p > 2
c‹‚‘y•‘‚ –• •ŽŒ•
y…ŽŒ†•„ •Ž›‘~ ‡~‘y €Ž~‰‰z• ‡~† –• •ŽŒ• ‘„Š y…ŽŒ†•„ •Ž›‘~ ‡~‘y
P‘„Š
+∞
1
m n
1
n
•‘{ˆ‚• ‘„ •†•ˆ{ •‚†Žy m,n=1 xm,n ™•Œ’ xm,n = m+1
( m+1
) − m+2
( m+1
m+2 ) €†~ ‡y…‚ m, n
P
xk
aŽy•‘‚ ‘„ •‚†Žy +∞
k=1 1+x2k –• •†•ˆ{ •‚†Žy ‰‚ ‘z‘Œ†Œ ‘Ž™•Œ ›•‘‚ Š~ ~•Œ•‚|‹‚‘‚ ™‘† †•”š‚†
P+∞ xk
P+∞ (−1)k−1 x2k−1
€†~ ‡y…‚ x ∈ (−1, 1)
k=1 1+x2k =
k=1
1−x2k−1
l‚‡†Š›Š‘~• ~•™ ‘„ •†•ˆ{ •‚†Žy
P+∞
m,n=1 mx
xm
m=1 m 1−xm
P+∞
=
mn
~•Œ•‚|‹‘‚ ™‘† €†~ ‡y…‚ x ‰‚ |x| < 1 †•”š‚†
xn
n=1 (1−xn )2 .
P+∞
P+∞
P+∞
m
n
X•‘–
n x ‡~† g(x) =
m=1 bm x €†~ ‡y…‚ x ∈ (−R, R) _•Œ•‚|‹‘‚ ™‘†
Pa+∞
P+∞f (x) =n n=1
m
†•”š‚† n=1 an g(x ) = m=1 bm f (x ) €†~ ‡y…‚ x ∈ (−R1 , R1 ) ™•Œ’ R1 = PLQ{R, 1}
a†Š™‰‚ŠŒ &DXFK\ •‚†Ž›Š
P+∞
P+∞
a
‡~†
mogpjmp
X•‘–
Œ†
•‚†Žz•
m
n=0 bn p”„‰~‘|ƒŒ’‰‚ ‘Œ’• ™ŽŒ’• ‰†~• Šz~•
m=0
P+∞
•‚†Žy• k=0 ck –• ‚‹{• c0 = a0 b0 c1 = a1 b0 + a0 b1 c2 = a2 b0 + a1 b1 + a0 b2 ‡~† €‚Š†‡™‘‚Ž~
ck =
e •‚†Žy
Pk
l=0 ak−l bl
= ak b0 + ak−1 b1 + · · · + a1 bk−1 + a0 bk
€†~ ‡y…‚ k ≥ 0.
P+∞ Pk
c
=
a
b
k
k−l
l
k=0
k=0
l=0
P+∞
P+∞
RŠŒ‰yƒ‚‘~† €†Š™‰‚ŠŒ &DXFK\ ‘–Š m=0 am ‡~† n=0 bn
P+∞
e †•z~ €†~
‚|•Œ’• •Œˆˆ~•ˆ~•†~•‰™ •ŽŒzŽ”‚‘~† ~•™ ‘†• •’Š~‰Œ•‚†Žz• •„ˆ~•{ •‚†Žz•
P‘z‘Œ†Œ’
+∞
m _’‘z• ‘†• •‚†Žz• …~ ‘†• ‰‚ˆ‚‘{•Œ’‰‚ •‘„Š ‚Š™‘„‘~
_Š •Œˆˆ~
‘„• ‰ŒŽ“{• m=0 am xP
P+∞
n ™•–• •Œˆˆ~•ˆ~•†yƒŒ’‰‚ •šŒ •Œˆ’›Š’‰~
m ‡~†
b
x
a
x
•ˆ~•†y•Œ’‰‚ ‘†• •‚†Žz• +∞
n
m
n=0
m=0
P
k
•„ˆ~•{ Œ‰~•Œ•Œ†›Š‘~• |•†‚• •’Šy‰‚†• ‘Œ’ x •ˆz•Œ’‰‚ ™‘† •”„‰~‘|ƒ‚‘~† „ •‚†Žy +∞
k=0 ck x ‘„•
Œ•Œ|~• Œ† •’Š‘‚ˆ‚•‘z• cn •|ŠŒŠ‘~† ~•™ ‘Œ’• •~Ž~•yŠ– ‘š•Œ’•
P
P+∞
fcvoej_
_Š Œ† •‚†Žz• +∞
m=0 am ‡~†
n=0 bn •’€‡ˆ|ŠŒ’Š ~•Œˆš‘–• ‘™‘‚ ‘Œ €†Š™‰‚ŠŒ
P+∞
&DXFK\ k=0 ck ‘–Š •šŒ •‚†Ž›Š •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~†
P+∞
P+∞
P+∞
n=0 bn .
m=0 am
k=0 ck =
P
_•™•‚†‹„ f‚–ŽŒš‰‚ ‘„ •†•ˆ{ •‚†Žy +∞
m,n=0 xm,n ŒŽ|ƒŒŠ‘~• xm,n = am bn €†~ ‡y…‚ m, n ≥ 0
e •†•ˆ{ •‚†Žy •’€‡ˆ|Š‚† ~•Œˆš‘–• •†™‘†
P+∞
P+∞
P+∞ P+∞
P+∞
P+∞
m=0
n=0 |xm,n | =
m=0 |am |
n=0 |bn | =
m=0 |am |
n=0 |bn | < +∞.
‡~† •Ž|•‡Œ’‰‚
c“~Ž‰™ƒŒ’‰‚ ‘Œ …‚›Ž„‰~
P+∞ P+∞
P+∞ Pk
k=0
m=0
l=0 xk−l,l =
n=0 xm,n .
p’Š‚•y€‚‘~†
P+∞
k=0 ck
=
P+∞ Pk
k=0
l=0 ak−l bl
=
P+∞
m=0 am
P+∞
n=0 bn .
P
qzˆŒ• ‚“~Ž‰™ƒŒŠ‘~• ‘Œ …‚›Ž„‰~
•‘„ •‚†Žy +∞
m,n=0 |xm,n | €†~ ‘„Š ‘‚ˆ‚’‘~|~ †•™‘„‘~ •~Ž~
‡y‘– •Ž|•‡Œ’‰‚
P+∞ Pk
P+∞
P+∞
P+∞
P+∞ Pk
k=0
l=0 |ak−l ||bl | =
m=0 |am |
n=0 |bn |
k=0 |ck | =
k=0
l=0 ak−l bl ≤
< +∞,
Œ•™‘‚ „
P+∞
k=0 ck
•’€‡ˆ|Š‚† ~•Œˆš‘–•
P
n
n~Žy•‚†€‰~
a†Š™‰‚ŠŒ &DXFK\ ‘„• •‚†Žy• +∞
n=0 a ‰‚ ‘ŒŠ ‚~’‘™ ‘„•
c|Š~† 1 · 1 = 1 ‡~†
ak 1 + ak−1 a + · · · + aak−1 + 1ak = (k + 1)ak
P
P
n
k
€†~ ‡y…‚ k VŽ~P
‘Œ €†Š™‰‚ŠŒ &DXFK\ ‘„• +∞
„ +∞
n=0 a ‰‚ ‘ŒŠ ‚~’‘™ ‘„• ‚|Š~†
k=0 (k + 1)a
P
+∞ n
+∞
k
_Š |a| < 1 „ n=0 a •’€‡ˆ|Š‚† ~•Œˆš‘–• ‡~† ‚•Œ‰zŠ–• ‡~† „ k=0 (k + 1)a •’€‡ˆ|Š‚†
~•Œˆš‘–• ‡~†
P+∞
P+∞ m P+∞ n
1
k
~Š |a| < 1.
k=0 (k + 1)a =
m=0 a
n=0 a = (1−a)2
P+∞
b‚|‘‚ ‘„Š
€†~ zŠ~Š •†~“ŒŽ‚‘†‡™ ‘Ž™•Œ ~•™•‚†‹„• ”–Ž|• ”Ž{•„ •†•ˆ›Š •‚†Ž›Š ‘Œ’
‘š•Œ’ P
k=0 ck =
P+∞
Py•‡„•„
P+∞
+∞
+∞
a
b
‡~†
‰yˆ†•‘~
‰‚
~•…‚Šz•‘‚Ž‚•
’•Œ…z•‚†•
‰™ŠŒ
„
‰|~
~•™
‘†•
•šŒ
•‚†Žz•
a
m
n
m
m=0
n=0
m=0
n=0 bn
”Ž‚†yƒ‚‘~† Š~ •’€‡ˆ|Š‚† ~•Œˆš‘–• ‚Š› „ yˆˆ„ •Žz•‚† ~•ˆ›• Š~ •’€‡ˆ|Š‚† _Š ‰|~ ~•™ ‘†• •šŒ •‚†Žz• •‚Š •’€‡ˆ|Š‚†
~•Œˆš‘–• •‚Š ‰•ŒŽŒš‰‚ Š~ •’‰•‚ŽyŠŒ’‰‚ ‚Š €zŠ‚† ™‘† ‘Œ €†Š™‰‚ŠŒ &DXFK\ •’€‡ˆ|Š‚† ~•Œˆš‘–• XŠ~ ~‡™‰„ •”‚‘†‡™
~•Œ‘zˆ‚•‰~ ‚|Š~† •‘„Š y•‡„•„
P+∞
P
am
n~Žy•‚†€‰~
aŠ–Ž|ƒŒ’‰‚ ™‘† Œ† •‚†Žz• +∞
n=0
m=0 m! ‡~†
‡y…‚ a, b
c|Š~† 1 · 1 = 1 ‡~† •y•‚† ‘Œ’ •†–Š’‰†‡Œš ‘š•Œ’ ‘Œ’ 1HZWRQ ‚|Š~†
ak
k! 1
+
ak−1 b1
(k−1)! 1!
+ ··· +
€†~ ‡y…‚ k VŽ~ ‘Œ €†Š™‰‚ŠŒ &DXFK\ ‘–Š
P+∞
k=0
a1 bk−1
1! (k−1)!
am
m=0 m!
P+∞
(a+b)k
k!
=
‡~†
am
m=0 m!
P+∞
bn
n!
•’€‡ˆ|ŠŒ’Š ~•Œˆš‘–• €†~
+ 1 bk! =
(a+b)k
k!
bn
n=0 n!
‚|Š~† „
k
P+∞
bn
n=0 n! .
P+∞
P+∞
k=0
(a+b)k
k!
Œ•™‘‚
m† •‚†Žz• ~’‘z• ‡~† Œ ‘š•Œ• •‘ŒŠ Œ•Œ|Œ ‡~‘~ˆ{‹~‰‚ •”‚‘|ƒŒŠ‘~† y‰‚•~ ‰‚ ‘„Š ‚‡…‚‘†‡{ •’ŠyŽ
‘„•„ b‚|‘‚ ‘„Š y•‡„•„
qŒ …z‰~ ~’‘™ …~ ‰‚ˆ‚‘„…‚| •†‚‹Œ•†‡y •‘~ •~Ž~•‚|€‰~‘~
‡~†
_•‡{•‚†•
_•Œ•‚|‹‘‚ ™‘†
P+∞
k=0
(k+1)(k+2) k
a
2
=
1
(1−a)3
€†~ ‡y…‚ a ∈ (−1, 1)
P
P+∞ (−1)n
1
_•Œ•‚|‹‘‚ ™‘† x1 + +∞
k=1 x(x+1)···(x+k) = e
n=0 n!(x+n) €†~ ‡y…‚ x 6= 0, −1, −2, . . .
”Ž„•†‰Œ•Œ†›Š‘~• ‘Œ ‘‚ˆ‚’‘~|Œ ~•Œ‘zˆ‚•‰~ ‘„• y•‡„•„•
P
xn
~•Œ•‚|‹~‰‚ ™‘† €†~ ‡y…‚ a, b
a†~ ‡y…‚ x ŒŽ|ƒŒ’‰‚ f (x) = +∞
n=0 n! p‘Œ •~Žy•‚†€‰~
†•”š‚† f (a + b) = f (a)f (b) c•|•„• €Š–Ž|ƒŒ’‰‚ ™‘† f (0) = 1 ‡~† f (1) = e
_•Œ•‚|‹‘‚ ™‘† „ f : R → R ‚|Š~† •’Š‚”{• •‘ŒŠ 0
>€@ ~•Œ•‚|‹‘‚ ™‘† †•”š‚† f (x) = ex €†~ ‡y…‚ x •„ˆ~•{ ™‘† †•”š‚†
`y•‚†
y•‡„•„•
P+∞ ‘„•
xn
x
‡~†
n=0 n! = e €†~ ‡y…‚ x _’‘™ …~ ‘Œ ‹~Š~~•Œ•‚|‹Œ’‰‚ •‘~ •~Ž~•‚|€‰~‘~
P+∞
P+∞
_•Œ•‚|‹‘‚ ‘Œ …‚›Ž„‰~ ‘Œ’ 0HUWHQV ~Š „ ‰|~ ~•™ ‘†• •‚†Žz•
P+∞ m=0 am ‡~† n=0 bn •’
€‡ˆ|Š‚†
~•Œˆš‘–•P
‡~† „ yˆˆ„ P
•’€‡ˆ|Š‚† ‘™‘‚ ‘Œ €†Š™‰‚ŠŒ &DXFK\ k=0 ck ‘–Š •šŒ •‚†Ž›Š •’€‡ˆ|Š‚†
P+∞
+∞
+∞
‡~† k=0 ck = m=0 am
n=0 bn
P+∞ (−1)n−1
e n=1 √n •’€‡ˆ|Š‚† _•Œ•‚|‹‘‚ ™‘† ‘Œ €†Š™‰‚ŠŒ &DXFK\ ‘„• •‚†Žy• ~’‘{• ‰‚ ‘ŒŠ ‚~’‘™ ‘„•
~•Œ‡ˆ|Š‚†
P
(−1)n−1
•’€‡ˆ|Š‚† _•Œ•‚|‹‘‚ ™‘† ‘Œ €†Š™‰‚ŠŒ &DXFK\ ‘„• •‚†Žy• ~’‘{•
aŠ–Ž|ƒŒ’‰‚ ™‘† „ +∞
n=1 P n
+∞
2
‰‚ ‘ŒŠ ‚~’‘™ ‘„• ‚|Š~† „ k=1 (−1)k−1 zk ™•Œ’ zk = k+1
(1 + 21 + · · · + k1 ) €†~ ‡y…‚ k ‡~† ™‘†
~’‘{ „ •‚†Žy •’€‡ˆ|Š‚† ~ˆˆy ™”† ~•Œˆš‘–•
P
P
(−1)n−1
1
‡~† +∞
f‚–Ž{•‘‚ ‘†• +∞
m=0 4m ‡~† ~•Œ•‚|‹‘‚ ”–Ž|• Š~ ‚“~Ž‰™•‚‘‚ ‘Œ …‚›Ž„‰~ ‘Œ’
n=1
n
0HUWHQV ™‘† ‘Œ €†Š™‰‚ŠŒ &DXFK\ ‘Œ’• •’€‡ˆ|Š‚† ~ˆˆy ™”† ~•Œˆš‘–•
_Š~•†~‘y‹‚†• •‚†Ž›Š
mogpjmp
f‚–ŽŒš‰‚ ‰†~ zŠ~ •ŽŒ• zŠ~ ‡~† ‚•| •’ŠyŽ‘„•„ σ : N → N _’‘™ •„‰~|Š‚† ™‘†
Œ† ~Ž†…‰Œ| σ(1) σ(2) σ(3), . . . •‚Ž†ˆ~‰•yŠŒ’Š ‡y…‚ “’•†‡™ ~Ž†…‰™ ~‡Ž†•›• ‰|~ “ŒŽy { ‰‚ yˆˆ~
ˆ™€†~ ~•Œ‘‚ˆŒšŠ ‰†~ ~Š~•†y‘~‹„ ‘–Š “’•†‡›Š ~Ž†…‰›Š
q›Ž~ z•‘– ~‡ŒˆŒ’…|~ (xn ) _Š •’‰•Œˆ|•Œ’‰‚ x′n = xσ(n) ‘™‘‚ „ ~‡ŒˆŒ’…|~ (x′n ) ŒŠŒ‰yƒ‚‘~† ~Š~
P
P+∞
′
•†y‘~‹„ ‘„• (xn ) c•|•„• ˆz‰‚ ™‘† „ •‚†Žy +∞
n=1 xn ‚|Š~† ‰†~ ~Š~•†y‘~‹„ ‘„• •‚†Žy•
n=1 xn
_Š
P
s′n = nk=1 x′k ,
P+∞
P
′
€†~ ‡y…‚ n ‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘–Š •‚†Ž›Š n=1
xn ‡~† +∞
n=1 xn ‘™‘‚ •‚Š ‰•ŒŽŒš‰‚
Š~ •‚Ž†‰zŠŒ’‰‚ Œ† ~Ž†…‰Œ| sn ‡~† s′n Š~ ‚|Š~† |•†Œ† qŒ y…ŽŒ†•‰~ sn •‚Ž†z”‚† ‘Œ’• x1 , x2 , . . . , xn
‚Š› ‘Œ s′n ‘Œ’• xσ(1) , xσ(2) , . . . , xσ(n) c•Œ‰zŠ–• •‚Š ‚|Š~† ‡~…™ˆŒ’ •z•~†Œ ‡~† ‚Š €zŠ‚† •‚Š
sn =
Pn
k=1 xk
‡~†
P
P+∞ ′
†•”š‚† ™‘† OLPn→+∞ sn = OLPn→+∞ s′n { †•Œ•šŠ~‰~ +∞
n=1 xn =
n=1 xn j•ŒŽ‚| „ ‰|~ •‚†Žy
Š~ •’€‡ˆ|Š‚† ‚Š› „ yˆˆ„ Š~ ~•Œ‡ˆ|Š‚† { Š~ •’€‡ˆ|ŠŒ’Š ‡~† Œ† •šŒ ~ˆˆy Š~ z”Œ’Š •†~“ŒŽ‚‘†‡y
~…ŽŒ|•‰~‘~
P
(−1)n−1
= 1 − 12 + 13 − 41 + · · · •’€‡ˆ|Š‚†
n~Žy•‚†€‰~
aŠ–Ž|ƒŒ’‰‚ ™‘† „ •‚†Žy +∞
n=1
n
1
1
1
1
1
1
1
1
q›Ž~ „ •‚†Žy 1 + 3 − 2 + 5 + 7 − 4 + 9 + 11 − 6 + · · · ‚|Š~† ~Š~•†y‘~‹„ ‘„• •ŽŒ„€Œš‰‚Š„•
f~ ~•Œ•‚|‹Œ’‰‚ ™‘† ‡~† „ •‚š‘‚Ž„ •‚†Žy •’€‡ˆ|Š‚† ~ˆˆy ™‘† z”‚† •†~“ŒŽ‚‘†‡™ y…ŽŒ†•‰~ ~•™ ‘„Š
•Ž›‘„
_Š sn ‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚š‘‚Ž„• •‚†Žy• ‘™‘‚
1
1
1
+ 4n−1
− 2n
.
s3n = 1 + 13 − 12 + 51 + 17 − 41 + · · · + 4n−3
VŽ~
1
4n+1
s3(n+1) − s3n =
+
1
4n+3
−
1
2n+2
‡~† ‚•Œ‰zŠ–• „ ~‡ŒˆŒ’…|~ (s3n ) ‚|Š~† ~š‹Œ’•~ c•|•„•
1
1
− 4n−3
−
s3n = 1 + 13 − 12 − 15 − 71 − · · · − 2n−2
>0
1
4n−1
−
1
2n
<1+
1
3
= 34 ,
‚•‚†•{ ‡y…‚ •~ŽzŠ…‚•„ ‚|Š~† …‚‘†‡{ VŽ~ „ ~‡ŒˆŒ’…|~ (s3n ) ‚|Š~† ‡~† yŠ– “Ž~€‰zŠ„ ‡~† ‚•Œ‰z
Š–• •’€‡ˆ|Š‚† •‚ ‡y•Œ†ŒŠ s q›Ž~
s3n+1 = s3n +
1
4n+1
→s+0=s
‡~† s3n+2 = s3n +
1
4n+1
+
1
4n+3
→ s + 0 + 0 = s.
VŽ~ sn → s ‡~† ‚•Œ‰zŠ–•
1
3
+
1
5
+
1
7
−
s3n ≥ 1 +
1
3
−
1
2
1+
−
1
2
1
4
+
1
9
+
1
11
−
1
6
+
1
5
+
1
7
−
1
4
=
1
3
−
1
4
+ ···
+ · · · = s.
n~Ž~‘„ŽŒš‰‚ ™‘†
€†~ ‡y…‚ n ≥ 2 Œ•™‘‚ s ≥
X•‘– ‘›Ž~
5
6
+
13
140
t=1−
1
2
+
5
6
+
13
140
‘Œ y…ŽŒ†•‰~ ‘„• ~Ž”†‡{• •‚†Žy• _Š tn ‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘y ‘„• ‘™‘‚
1
1
+ 2n−1
= 1 − 21 + 31 − 14 − 15 − · · · −
t2n−1 = 1 − 12 + · · · − 2n−2
≤1−
1
2
€†~ ‡y…‚ n VŽ~ t ≤
+
1
3
=
5
6
1
2n−2
−
1
2n−1
5
6
‡~† ‚•Œ‰zŠ–• t < s
P
fcvoej_
X•‘– ™‘† „ •‚†Žy +∞
~•Œˆš‘–• q™‘‚
Œ•Œ†~•{•Œ‘‚ ~Š~•†y‘~‹„
n=1 xn •’€‡ˆ|Š‚† P
P+∞
P+∞
P+∞ ′
+∞ ′
=
‘„•
x
•’€‡ˆ|Š‚†
‚•|•„•
~•Œˆš‘–•
‡~†
x
x
n=1 xn
n=1 n
n=1 n
n=1 n
nŽ›‘„ ~•™•‚†‹„ X•‘– σ : N → N „ zŠ~ •ŽŒ• zŠ~ ‡~† ‚•| •’ŠyŽ‘„•„ „ Œ•Œ|~ ŒŽ|ƒ‚† ‘„Š ~Š~
•†y‘~‹„ b„ˆ~•{ x′n = xσ(n) €†~ ‡y…‚ n
mŽ|ƒŒ’‰‚
(
xn = xσ(m) = x′m , ~Š n = σ(m)
xm,n =
0,
~Š n 6= σ(m)
P+∞
€†~ ‡y…‚ m, n nŽŒ‡š•‘‚† „ •†•ˆ{ •‚†Žy m,n=1 xm,n „ Œ•Œ|~ •’€‡ˆ|Š‚† ~•Œˆš‘–• •†™‘†
P+∞
P+∞ P+∞
n=1
m=1 |xm,n | =
n=1 |xn | < +∞.
_•™ ‘Œ …‚›Ž„‰~
•’Š‚•y€‚‘~†
P+∞ P+∞
m=1
n=1 xm,n
=
P+∞ P+∞
n=1
m=1 xm,n
,
P
P+∞
′
Œ•™‘‚ +∞
m=1 xm =
n=1 xn
P+∞
b‚š‘‚Ž„
n=1 |xn | Œ•™‘‚ 0 ≤ S < +∞
P+∞~•™•‚†‹„ fz‘Œ’‰‚ S =P+∞
P
e n=1 xn •’€‡ˆ|Š‚† ‡~† z•‘– n=1 xn = s f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = nk=1 xk
Œ•™‘‚ sn → s
P
n~Ž~‘„ŽŒš‰‚ ™‘† †•”š‚† nk=1 |x′k | ≤ S €†~ ‡y…‚ n ~“Œš Œ† ™ŽŒ† |x′1 |, . . . , |x′n | •„ˆ~•{ Œ†
™ŽŒ† |x
|, . . . , |xσ(n) | ‚|Š~† ‡y•Œ†Œ† ~•™ ‘Œ’• |x1 |, |x2 |, . . . ”–Ž|• ‚•~Šyˆ„•„ VŽ~ „ •‚†Žy
P+∞ σ(1)
′ | •’€‡ˆ|Š‚†
|x
n=1 n
Pn
′
′
′
f‚–ŽŒš‰‚
P+∞ ′ ‡~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = k=1 xk ‡~† …~ ~•Œ•‚|‹Œ’‰‚ ™‘† sn → s •„ˆ~•{ ™‘†
~•™•‚†‹„ …~ z”‚† ‘‚ˆ‚†›•‚†
n=1 xn = s Œ•™‘‚ „P
X•‘– ǫ > 0 c•‚†•{ „ +∞
n=1 |xn | •’€‡ˆ|Š‚† ‡~† ‚•‚†•{ sn → s ’•yŽ”‚† n0 ›•‘‚ Š~ †•”š‚†
P+∞
ǫ
‡~† |sn − s| < 2ǫ
€†~ ‡y…‚ n ≥ n0 .
n=n0 +1 |xn | < 2
c•†ˆz€Œ’‰‚ n1 ~Ž‡‚‘y ‰‚€yˆŒ ›•‘‚ Œ† σ(1), . . . , σ(n1 ) Š~ •‚Ž†ˆ~‰•yŠŒ’Š ‘Œ’• 1, 2, . . . , n0
c|Š~† •ŽŒ“~Šz• ™‘† n1 ≥ n0 _Š n ≥ n1 (≥ n0 ) ‘™‘‚ •‘Œ s′n − sn •‚Š •‚Ž†ˆ~‰•yŠŒŠ‘~† Œ†
x1 , . . . , xn0 ~“Œš ‡~…zŠ~• ~•™ ~’‘Œš• •‚Ž†z”‚‘~† ~‡Ž†•›• ‰|~ “ŒŽy •‘Œ sn ‡~† •‘Œ s′n c•Œ‰z
Š–• ~Š n ≥ n1 ~•™ ‘„Š
•’Š‚•y€‚‘~†
P
ǫ
|s′n − sn | ≤ +∞
n=n0 +1 |xn | < 2 .
_•™ ‘„Š ‘‚ˆ‚’‘~|~ •”z•„ ‡~† •yˆ† ~•™ ‘„Š
•’Š‚•y€‚‘~† ™‘† €†~ ‡y…‚ n ≥ n1 †•”š‚†
|s′n − s| ≤ |s′n − sn | + |sn − s| <
ǫ
2
+
ǫ
2
= ǫ.
VŽ~ s′n → s
e š•~Ž‹„ ‘Œ’ •ŽŒ„€Œš‰‚ŠŒ’ •~Ž~•‚|€‰~‘Œ•
z”‚† –• •~…š‘‚Ž„ ~†‘|~ ‘Œ ™‘† „ ~Ž”†‡{ •‚†Žy
•’€‡ˆ|Š‚† ~ˆˆy ™”† ~•Œˆš‘–• _’‘™ …~ “~Š‚| ‡~† ~•™ ‘Œ …‚›Ž„‰~ ‘Œ’ 5LHPDQQ •Œ’ ~‡ŒˆŒ’…‚|
l e ~•™•‚†‹„ ‘Œ’ …‚–Ž{‰~‘Œ• ‘Œ’ 5LHPDQQ •’Š~€–Š|ƒ‚‘~† •‚ •’•‡Œˆ|~ ‘„Š ~•™•‚†‹„ ‘Œ’
…‚–Ž{‰~‘Œ•
P
fcvoej_ qmr 5,(0$11 X•‘–P
™‘† „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ’•™ •’Š…{‡„ q™‘‚ €†~ ‡y…‚
+∞ ′
a, b ∈ R ‰‚ a ≤ b ’•yŽ”‚† ~Š~•†y‘~‹„ n=1 xn ‘„• ~Ž”†‡{• •‚†Žy• ›•‘‚ ~Š s′n = x′1 + · · · + x′n
‚|Š~† ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ ‘„• •‚š‘‚Ž„• •‚†Žy• Š~ ‚|Š~†
OLP s′n = a,
OLP s′n = b.
P+∞
P+∞
P
_•™•‚†‹„ X”Œ’‰‚ ™‘† „ +∞
n=1 xn
n=1 |xn | = +∞ X•‘– s =
n=1 xn •’€‡ˆ|Š‚† ~ˆˆy
f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ sn = x1 + · · · + xn ‡~† Sn = |x1 | + · · · + |xn |
_•™ ‘Œ’• xn ŒŽ|ƒŒ’‰‚ y1 Š~ ‚|Š~† Œ •Ž›‘Œ• Œ Œ•Œ|Œ• ‚|Š~† ≥ 0 y2 Œ •‚š‘‚ŽŒ• Œ Œ•Œ|Œ• ‚|Š~† ≥ 0
y3 Œ ‘Ž|‘Œ• Œ Œ•Œ|Œ• ‚|Š~† ≥ 0 ‡~† Œš‘– ‡~… ‚‹{• m‰Œ|–• ~•™ ‘Œ’• xn ŒŽ|ƒŒ’‰‚ z1 Š~ ‚|Š~† Œ
•Ž›‘Œ• Œ Œ•Œ|Œ• ‚|Š~† < 0 z2 Œ •‚š‘‚ŽŒ• Œ Œ•Œ|Œ• ‚|Š~† < 0 z3 Œ ‘Ž|‘Œ• Œ Œ•Œ|Œ• ‚|Š~† < 0 ‡~†
Œš‘– ‡~… ‚‹{•
f~ ~•Œ•‚|‹Œ’‰‚ ™‘†
P+∞
P+∞
n=1 yn = +∞,
n=1 zn = −∞.
f‚–ŽŒš‰‚ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ tn = y1 + · · · + yn ‡~† un = z1 + · · · + zn
P
k|
n
n~Ž~‘„ŽŒš‰‚ ™‘† Œ sn +S
= nk=1 xk +|x
‚|Š~† |•Œ• ‰‚ ‘Œ y…ŽŒ†•‰~ ‘–Š ‰„ ~ŽŠ„‘†‡›Š ~•™ ‘Œ’•
2
2
sn +Sn
n
≤ tn c•‚†•{ sn +S
→ s+∞
x1 , . . . , xn ‡~† ‚•Œ‰zŠ–•
2
2
2 = +∞ •’Š‚•y€‚‘~† tn → +∞
P+∞
VŽ~ n=1 yn = +∞
P
k|
n
= k=1 xk −|x
‚|Š~† |•Œ• ‰‚ ‘Œ y…ŽŒ†•‰~ ‘–Š ~ŽŠ„‘†‡›Š ~•™ ‘Œ’• x1 , . . . , xn
m‰Œ|–• Œ sn −S
2
2
P+∞
sn −Sn
sn −Sn
Œ•™‘‚ 2 ≥ un c•‚†•{ 2 → s−∞
n=1 zn = −∞
2 = −∞ z”Œ’‰‚ un → −∞ VŽ~
f‚–ŽŒš‰‚ •šŒ •’€‡‚‡Ž†‰zŠ‚• ~‡ŒˆŒ’…|‚• (an ) (bn ) ›•‘‚
an → a,
bn → b
–• ‚‹{• mŽ|ƒŒ’‰‚ an = a ~Š a ∈ R ‡~† an = −n ~Š a = −∞ ‡~† an = n ~Š a = +∞ qŒ
|•†Œ ~‡Ž†•›• ‡yŠŒ’‰‚
‡~† ‰‚ ‘Œ b n~Ž~‘„Ž{•‘‚ ™‘† †•”š‚† ‘‚ˆ†‡y an ≤ bn
P
`{‰~ c•‚†•{ +∞
n=1 yn = +∞ ’•yŽ”‚† n1 ›•‘‚ y1 + · · · + yn1 > b1 ‡~† z•‘– ™‘† Œ n1 ‚|Š~† Œ
‚ˆy”†•‘Œ• ‰‚ ~’‘{Š ‘„Š †•†™‘„‘~ •„ˆ~•{ ™‘† y1 + · · · + yn1 −1 ≤ b1 c•Œ‰zŠ–•
b1 < y1 + · · · + yn1 ≤ b1 + yn1 .
P+∞
c•‚†•{ n=1 zn = −∞ ’•yŽ”‚† n1 ∗ ›•‘‚ z1 + · · · + zn1 ∗ < a1 − (y1 + · · · + yn1 ) ‡~† z•‘– ™‘† Œ
n1 ∗ ‚|Š~† Œ ‚ˆy”†•‘Œ• ‰‚ ~’‘{Š ‘„Š †•†™‘„‘~ •„ˆ~•{ ™‘† z1 + · · · + zn1 ∗ −1 ≥ a1 − (y1 + · · · + yn1 )
c•Œ‰zŠ–•
a1 + zn1 ∗ ≤ y1 + · · · + yn1 + z1 + · · · + zn1 ∗ < a1 .
P+∞
`{‰~ c•‚†•{ n=n1 +1 yn = +∞ ’•yŽ”‚† n2 > n1 ›•‘‚ yn1 +1 + · · · + yn2 > b2 − (y1 +
· · · + yn1 + z1 + · · · + zn1 ∗ ) ‡~† z•‘– ™‘† Œ n2 ‚|Š~† Œ ‚ˆy”†•‘Œ• ‰‚ ~’‘{Š ‘„Š †•†™‘„‘~ •„ˆ~•{
™‘† yn1 +1 + · · · + yn2 −1 ≤ b2 − (y1 + · · · + yn1 + z1 + · · · + zn1 ∗ ) c•Œ‰zŠ–•
b2 < y1 + · · · + yn1 + z1 + · · · + zn1 ∗ + yn1 +1 + · · · + yn2 ≤ b2 + yn2 .
P
∗
∗
c•‚†•{ +∞
n=n1 ∗ +1 zn = −∞ ’•yŽ”‚† n2 > n1 ›•‘‚ zn1 ∗ +1 +· · ·+zn2 ∗ < a1 −(y1 +· · ·+yn1 +
z1 + · · · + zn1 ∗ + yn1 +1 + · · · + yn2 ) ‡~† z•‘– ™‘† Œ n2 ∗ ‚|Š~† Œ ‚ˆy”†•‘Œ• ‰‚ ~’‘{Š ‘„Š †•†™‘„‘~
•„ˆ~•{ ™‘† zn1 ∗ +1 + · · · + zn2 ∗ −1 ≥ a1 − (y1 + · · · + yn1 + z1 + · · · + zn1 ∗ + yn1 +1 + · · · + yn2 )
c•Œ‰zŠ–•
a2 + zn2 ∗ ≤ y1 + · · · + yn1 + z1 + · · · + zn1 ∗ + yn1 +1 + · · · + yn2 + zn1 ∗ +1 + · · · + zn2 ∗ < a2 .
p’Š‚”|ƒŒ’‰‚ ‚• y•‚†ŽŒŠ ‚•†ˆz€ŒŠ‘~• •†~•Œ”†‡y ‘Œ’• y1 , . . . , yn1 z1 , . . . , zn1 ∗ yn1 +1 , . . . , yn2
zn1 ∗ +1 , . . . , zn2 ∗ , . . . Œ† Œ•Œ|Œ† •‘„ •‚†Žy ~’‘{ •Œ’ ‚‰“~Š|ƒŒŠ‘~† •‚Š ‚|Š~† ‘|•Œ‘‚ yˆˆŒ ~•™ ‰†~
~Š~•†y‘~‹„ ‘–Š xn f‚–ŽŒš‰‚ ‘›Ž~ ‘~ ‰‚Ž†‡y ~…ŽŒ|•‰~‘~ s′n ‘„• •’€‡‚‡Ž†‰zŠ„• ~Š~•†y‘~‹„•
m† •~Ž~•yŠ– •”z•‚†• •Œ’ †•”šŒ’Š •‚ ‡y…‚ •{‰~ ˆzŠ‚ ™‘† b1 < s′n1 ≤ b1 + yn1 a1 + zn1 ∗ ≤
s′n1 +n1 ∗ < a1 b2 < s′n1 +n1 ∗ +n2 ≤ b2 + yn2 a2 + zn2 ∗ ≤ s′n1 +n1 ∗ +n2 +n2 ∗ < a2 ‡~† €‚Š†‡™‘‚Ž~
ak + znk ∗ ≤ s′n1 +n1 ∗ +···+nk +nk ∗ < ak .
bk < s′n1 +n1 ∗ +···+nk ≤ bk + ynk ,
P
c•‚†•{ „ •‚†Žy +∞
n=1 xn •’€‡ˆ|Š‚† ‚|Š~† xn → 0 ‡~† ‚•‚†•{ Œ† (yn ) ‡~† (zn ) ‚|Š~† ’•Œ~‡ŒˆŒ’…|‚•
•’Š‚•y€‚‘~†
‘„• (xn ) •’Š‚•y€‚‘~† ynk → 0 ‡~† znk ∗ → 0 VŽ~ ~•™ ‘†•
s′n1 +n1 ∗ +···+nk +nk ∗ → a,
s′n1 +n1 ∗ +···+nk → b
‡~† ‚•Œ‰zŠ–•
OLP s′n ≤ a,
b ≤ OLP s′n .
c|Š~† •~“z• ™‘† €†~ ‡y…‚ n ≥ n1 ’•yŽ”‚† ‰ŒŠ~•†‡™• k ›•‘‚ ‚|‘‚ n1 + n1 ∗ + · · · + nk ≤ n <
n1 + n1 ∗ + · · · + nk + nk ∗ ‚|‘‚ n1 + n1 ∗ + · · · + nk + nk ∗ ≤ n < n1 + n1 ∗ + · · · + nk + nk ∗ + nk+1
p‘„Š •Ž›‘„ •‚Ž|•‘–•„ ‚|Š~†
s′n1 +n1 ∗ +···+nk +nk ∗ ≤ s′n ≤ s′n1 +n1 ∗ +···+nk
‡~† •‘„ •‚š‘‚Ž„ •‚Ž|•‘–•„ ‚|Š~†
s′n1 +n1 ∗ +···+nk +nk ∗ ≤ s′n ≤ s′n1 +n1 ∗ +···+nk +nk ∗ +nk+1 ,
Œ•™‘‚ ~•™ ‘†•
•’Š‚•y€‚‘~† ~Š‘†•‘Œ|”–•
ak + znk ∗ ≤ s′n ≤ bk + ynk
{
ak + znk ∗ ≤ s′n ≤ bk+1 + ynk+1 .
p’Š‚•y€‚‘~†
a = OLPk→+∞ (ak + znk ∗ ) ≤ OLP s′n ,
_•™ ‘†•
‡~†
OLP s′n ≤ OLPk→+∞ (bk + ynk ) = b.
z”Œ’‰‚ OLP s′n = a ‡~† OLP s′n = b
_•‡{•‚†•
1
27
r•ŒˆŒ€|•‘‚ ‘Œ y…ŽŒ†•‰~ ‘„• •‚†Žy• 1 +
+ 2115 + 2114 + 2113 + 2112 + 2111 + · · ·
1
2
+
1
23
+
1
22
+
1
26
+
1
25
+
1
24
+
1
210
+
1
29
+
1
28
+
X•‘– „ •’€‡ˆ|ŠŒ’•~ •‚†Žy 1 − 12 + 13 − 41 + 15 − 16 + · · · f‚–ŽŒš‰‚ ‘„Š ~Š~•†y‘~‹„
1
1
1
1
1
1
1
1
1
1
1
1
1− 12 + 31 − 14 + 15 + 71 − 61 + 19 + 11
+ 13
+ 15
− 81 + 17
+ 19
+ 21
+ 23
+ 25
+ 27
+ 29
+ 31
− 10
+· · ·
_•Œ•‚|‹‘‚ ™‘† „ •‚š‘‚Ž„ •‚†Žy ~•Œ‡ˆ|Š‚† •‘Œ +∞
P
X•‘– ™‘† „ +∞
n=1 xn •’€‡ˆ|Š‚† ’•™ •’Š…{‡„
P_•Œ•‚|‹‘‚ ™‘† €†~ ‡y…‚ s ’•yŽ”‚† ~‡ŒˆŒ’…|~
(ǫn ) ›•‘‚ Š~ †•”š‚† ǫn = ±1 €†~ ‡y…‚ n ‡~† ›•‘‚ +∞
n=1 ǫn xn = s
P
(−1)n−1
‡~† ‘{Š ~Š~•†~‘y••Œ’‰‚ •y•‚† ‘–Š ‚‹{•
X•‘– 0 < p < +∞ f‚–ŽŒš‰‚ ‘„Š +∞
n=1
n
•šŒ ‡~Š™Š–Š
(i) •‚Š ~ˆˆyƒŒ’‰‚ ‘„ •†y‘~‹„ ~Šy‰‚•~ •‘Œ’• …‚‘†‡Œš• ™ŽŒ’• Œš‘‚ ‘„ •†y‘~‹„ ~Šy‰‚•~ •‘Œ’•
~ŽŠ„‘†‡Œš• ™ŽŒ’•
(ii) ~Š ~•™ ‘Œ’• ~Ž”†‡Œš• n ™ŽŒ’• ‘„• •ŽŒ‡š•‘Œ’•~• •‚†Žy• Œ† kn ‚|Š~† …‚‘†‡Œ| ‡~† Œ† ln ‚|Š~†
~ŽŠ„‘†‡Œ| Œ•™‘‚ kn + ln = n ‘™‘‚ klnn → p
_•Œ•‚|‹‘‚ ™‘† „ •ŽŒ‡š•‘Œ’•~ •‚†Žy z”‚† y…ŽŒ†•‰~
1
2
ORJ(4p)
X•‘– •’Š~Ž‘{•‚†• fk : N → N €†~ ‡y…‚ k r•Œ…z‘Œ’‰‚ S
™‘† €†~ ‡y…‚ k „ fk ‚|Š~† zŠ~
•ŽŒ• zŠ~ ™‘† fk (N) ∩ fk′ (N) = ∅ €†~ ‡y…‚ k, k ′ ‰‚ k 6= k ′ ‡~† ™‘† +∞
k=1 fk (N) = N
P+∞
a†~ ‡y…‚ •‚†Žy n=1 xn „ Œ•Œ|~ •’€‡ˆ|Š‚† ~•Œˆš‘–• ~•Œ•‚|‹‘‚ ™‘†
P
(i) €†~ ‡y…‚ k „ ~Š‘|•‘Œ†”„ •‚†Žy +∞
m=1 xfk (m) •’€‡ˆ|Š‚† ~•Œˆš‘–•
P
P+∞
(ii) ~Š ŒŽ|•Œ’‰‚ sk = m=1 xfk (m) ‘™‘‚ „ •‚†Žy +∞
k=1 sk •’€‡ˆ|Š‚† ~•Œˆš‘–•
P+∞
P+∞
(iii) ~Š s = n=1 xn ‘™‘‚ k=1 sk = s b„ˆ~•{
P+∞
P+∞ P+∞
k=1
m=1 xfk (m) =
n=1 xn .
c•› •‚Ž†€Žy“‚‘~† Œ ‰„”~Š†•‰™• ‘„• ~Šyˆ’•„• •‚†Ž›Š •‚ ’•Œ•‚†Žz•
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