Math 523H–Homework 2
1. Let {tn } be a bounded sequence and {sn } be a convergent sequence with limn→∞ sn =
0. Show that limn→∞ (sn tn ) = 0.
2. Give a detailed proof of the following
Theorem For any sequence {sn } with sn > 0 we have limn→∞ sn = +∞ if and only
if limn→∞ s1n = 0.
3. (a) As proved in class if limn→∞ sn = +∞ and limn→∞ tn = t > 0 then limn→∞ (sn tn ) =
+∞. Construct examples of sequences {sn } and {tn } with limn→∞ sn = +∞
and limn→∞ tn = 0 and
i. limn→∞ (sn tn ) = +∞
ii. limn→∞ (sn tn ) = c for any arbitrary constant c.
iii. The sequence sn tn is bounded but not convergent.
(b) Suppose that limn→∞ sn = +∞ and tn is a bounded sequence. Show that
limn→∞ (sn + tn ) = +∞.
sn+1 = L exists.
4. Suppose {sn } is a sequence such that lim n→∞
sn (a) Show that if L < 1 then limn→∞ sn = 0.
Hint: Pick b such that L < b < 1 and choose N so large that sn+1
<b<1
sn for all n ≥ N . Then show that for n ≥ N we have |sn | ≤ bn−N |sN |.
(b) Show that if L > 1 then limn→∞ |sn | = ∞.
Hint: Proceed as in (a) or use Problem 2.
5. Use Problem 4 to prove that
(a) Let p > 0. Show that


an
lim p =
n→∞ n

0
+∞
does not exists
if |a| ≤ 1
if a > 1
.
if a < −1
an
= 0.
n→∞ n!
(b) Show that for any number a we have lim
6. Use Problems 3 and 5 to explain why
n4 + n
= +∞.
n→∞ n2 − 5
(a) lim
1
2n
+ (−5)n = +∞
n→∞ n10 + 1
10n 7n
(c) lim
− 3 = −∞
n→∞ n!
n
(b) lim
7. (a) Suppose {sn } is a sequence such that for all n we have
|sn − sn+1 | ≤ αn
for some α < 1. Then prove that {sn } is a Cauchy sequence.
1 − αk
to bound |sn − sn+k |.
Hint: Use the geometric series 1 + α + αk−1 =
1−α
(b) Suppose we are given a decimal expansion k.d1 d2 d3 · · · of a real number where
k is an integer and di ∈ {0, 1, 2, · · · , 9}. Use part (a) to show that
sn = k +
d1
dn
+ ··· + n
10
10
is a Cauchy sequence.
8. Consider the sequence
sn =
1
1
1
1
1
+
+
+
+ ··· +
1 · 5 3 · 7 5 · 9 7 · 11
(2n − 1)(2n + 3)
Show that {sn } is a Cauchy sequence and compute its limit.
Hint: Compute the partial fraction expansion of the
2
1
.
(2j − 1)(2j + 3)