Lecture 21Section 11.1 Infinite Series
Jiwen He
1
Infinite Series
What is an Infinite Series?
1
, · · · . Form the partial sums
Let the sequence ak = 21k : 21 , 14 , 18 , 16
1
X
1
s1 = a1 =
ak =
2
k=1
2
X
1 1
3
s2 = a1 + a2 =
ak = + =
2 4
4
k=1
3
X
7
1 1 1
s3 = a1 + a2 + a3 =
ak = + + =
2 4 8
8
k=1
4
X
1 1 1
1
15
s4 = a1 + a2 + a3 + a4 =
ak = + + +
=
2
4
8
16
16
..
k=1
n
. X
1
1 1 1
1
1
sn =
ak = + + + + + · · · + n = 1 − n+1
2 4 8
16
2
2
.. k=1
. X
∞
n
X
1
s∞ =
ak = lim sn = lim
ak = lim 1 − n+1 = 1
n→∞
n→∞
n→∞
2
k=1
The “infinite sum”
k=1
P∞
k=1
ak , is called infinite series.
Quiz
Quiz
1.
2.
Z
1
Z
∞
1
dx =
2
x
0
(a) 1, (b) 2, (c) ∞.
1
dx =
2
x
1
(a) 1, (b) 2, (c) ∞.
1
1.1
Basic Properties
Infinite Series: Definition
Given a sequence {ak }∞
k=1 , the infinite series is defined by the limit of the se∞
n
quence of partial sums:
X
X
ak = lim
ak
n→∞
k=1
k=1
The series converges if the limit exists; otherwise, it diverges.
∞
X
1 1 1
1
1
1
= + + +
+
+ ··· = 1
2k
2 4 8 16 32
k=1
Remarks
The summation index is a “dummy” index, much like the integration variable in
∞
∞
∞
X
X
X
a integral
1
1
1
=
=
k
i
2
2
2n
n=1
i=1
k=1
The starting value of the summation index needn’t be 1 and may be chosen for
∞
∞
∞
X
X
X
1
1
1
convenience
=
=
k
i+1
n−1
2 ∞i=0 2
2
∞
∞ n=2
k=1
X
X
X
1
1
1
1
=
=
−
=1
(k + 1)(k + 2)
i(i + 1)
i
i+1
i=1
i=1
k=0
sn =
1
1
1
1 1 1 1
− + − + ··· + −
=1−
→ 1 as n → ∞
1 2 2 3
i
i+1
i+1
Basic Test for Convergence
∞
X
If
ak converges, then ak → 0 as k → ∞
k=1
If ak 9 0 as k → ∞, then
∞
X
ak diverges.
k=1
Example 1.
∞
X
k
k
1 2
→ 1 6= 0 as k → ∞, then
= + + · · · diverges.
k+1
k+1
2 3
k=1
Remark
There are divergent series for which ak → 0:
∞
X
1
1
1 1
→ 0 as k → ∞, but
= 1 + + + · · · diverges.
k
k
2 3
k=1
2
Basic Properties
∞
∞
X
X
• If
ak converges and c is a constant, then
c ak converges, and
k=1
∞
X
c ak = c
k=1
• If
∞
X
ak and
k=1
∞
X
k=1
k=1
ak .
k=1
∞
X
bk both converge, then
∞
X
(ak + bk ) =
k=1
∞
X
∞
X
∞
X
ak +
k=1
(αak + βbk ) = α
k=1
∞
X
ak + β
k=1
(ak + bk ) converges, and
k=1
∞
X
bk .
k=1
∞
X
bk ,
∀α, β ∈ R
k=1
Quiz
Quiz
3.
4.
1.2
Z
1
Z
∞
1
dx =
0 x
(a) 1, (b) 2, (c) ∞.
1
dx =
x
1
(a) 1, (b) 2, (c) ∞.
Geometric Series
Geometric Series P
∞
k
Geometric Series:
k=0 x 

∞

X
k
x =


k=0
1
,
1−x
diverges,
if |x| < 1,
if |x| ≥ 1.
Proof.
• We have a “closed” formula for the nth partial sum:
1 − xn+1
sn = 1 + x + x2 + · · · + xn =
, if x 6= 1
1−x
• If |x| < 1, then xn+1 → 0 as n → ∞, thus sn →
1
1−x .
• If |x| > 1, then xn+1 is unbounded, thus sn diverges.
• If x = 1, then sn = n + 1 → ∞ as n → ∞
• If x = −1, then sn alternates between 0 and 1, thus diverges.
3
Examples
∞
∞
X
X
1
1
1 1 1
1
=
+
+
+
·
·
·
=
−1=
−1=1
k
k
2
2 4 8
2
1 − 12
k=1
k=0
k
∞
∞
X
X
1
2
(−1)k
1 1 1
1
=
=
=
1
−
+
−
+
·
·
·
=
−
2k
2 4 8
2
3
1 − − 12
k=0
k=0
∞
X
∞
X
1
1
1
1
= 0.1111111 · · · =
−1=
1 −1= 9
10k
10k
1
−
10
k=1
k=0
∞
∞
X
X
9
1
1
=
0.9999999
·
·
·
=
9
=9× =1
k
k
10
10
9
k=1
k=1
∞
X
ak
⇒ ak ∈ {0, · · · , 9},
= 0.a1 a2 a3 a4 a5 · · · = infinite decimal
10k
k=1
Examples
Suppose that a ball dropped from a height h hits the floor and rebounds to a
height σh with σ < 1, and so on. Find the total distance traveled by the ball if
h is 6 feet and σ = 32 .
4
The total distance traveled by the ball is
D = h + 2σh + 2σ 2 h + 2σ 3 h + · · ·
= h + 2σh 1 + σ + σ 2 + · · ·
= h + 2σh
∞
X
σk
k=0
2
1
1
=6+2× ×6
= h + 2σh
1−σ
3
1−
2
3
= 36 feet
Examples
∞
∞
X
7 23
73
3k+1 X 2k
+
+
+
·
·
·
=
−
5k
5 25 125
5k
5k
k=0
k=0
k=0
∞ k
∞ k
X
X
3
2
1
1
35
=3
−
=3
3 −
2 = 6
5
5
1
−
1
−
5
5
k=0
k=0
∞
∞
X
X
1
1
1 1
1
1
1
=
− 1+ +
+
+
+ ··· =
3k
27 81 243
3k
3 9
∞
X
3k+1 − 2k
=2+
k=3
k=0
1
=
1−
1
3
1
13
=
−
9
18
∞
∞
∞
X
X
1
1 X 1
1
1
1
1
=
=
+
+
+
·
·
·
=
3k
27 81 243
3i+3
27
3k
i=0
k=3
k=0
1
1
=
27 1 −
1
3
1
=
18
Outline
Contents
1 Infinite Series
1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1
2
3