§12.2–Infinite series
Mark Woodard
Furman U
Fall 2010
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
1 / 16
Outline
1
Some problems to set the stage
2
The proper definition
3
The geometric series
4
Telescoping series
5
The harmonic series
6
A test for divergence
7
Some useful theorems on series
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
2 / 16
Some problems to set the stage
Problem
Evaluate the following infinite sums:
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
3 / 16
Some problems to set the stage
Problem
Evaluate the following infinite sums:
1 1 1
S = 1 + + + + ···
2 4 8
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
3 / 16
Some problems to set the stage
Problem
Evaluate the following infinite sums:
1 1 1
S = 1 + + + + ···
2 4 8
S = 1 + 2 + 4 + 8 + ···
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
3 / 16
Some problems to set the stage
Problem
Evaluate the following infinite sums:
1 1 1
S = 1 + + + + ···
2 4 8
S = 1 + 2 + 4 + 8 + ···
S = 1 + (−1) + 1 + (−1) + · · ·
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
3 / 16
Some problems to set the stage
Problem
Evaluate the following infinite sums:
1 1 1
S = 1 + + + + ···
2 4 8
S = 1 + 2 + 4 + 8 + ···
S = 1 + (−1) + 1 + (−1) + · · ·
Commentary
The preceding examples underscore the need for a proper definition of the
sum of an infinite series.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
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The proper definition
Definition
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
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The proper definition
Definition
Given a sequence {an } of real numbers, we form a new sequence as
follows:
s1 = a1
s2 = a1 + a2
s3 = a1 + a2 + a3
..
.
The sequence {sn } is called the sequence of partial sums.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
4 / 16
The proper definition
Definition
Given a sequence {an } of real numbers, we form a new sequence as
follows:
s1 = a1
s2 = a1 + a2
s3 = a1 + a2 + a3
..
.
The sequence {sn } is called the sequence of partial sums.
P
If sn → L, then we say that the infinite series ∞
k=1 ak converges and
we write
∞
X
ak = L.
k=1
If {sn } diverges, then we say that the infinite series
Mark Woodard (Furman U)
§12.2–Infinite series
P∞
k=1 ak
diverges.
Fall 2010
4 / 16
The proper definition
Definition
P
A series
an is a collection of two sequences: the sequence of terms
{an } and the sequence of partial sums {sn } where sn = a1 + a2 + · · · + an .
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
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The proper definition
Problem
Determine whether or not the series
Mark Woodard (Furman U)
P∞
k+1
k=1 (−1)
§12.2–Infinite series
converges.
Fall 2010
6 / 16
The proper definition
Problem
Determine whether or not the series
P∞
k+1
k=1 (−1)
converges.
Solution
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
6 / 16
The proper definition
Problem
Determine whether or not the series
P∞
k+1
k=1 (−1)
converges.
Solution
Let sn denote the nth partial sum of the series. It is easy to see that
sn = 1 if n is odd and sn = 0 if n is even.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
6 / 16
The proper definition
Problem
Determine whether or not the series
P∞
k+1
k=1 (−1)
converges.
Solution
Let sn denote the nth partial sum of the series. It is easy to see that
sn = 1 if n is odd and sn = 0 if n is even.
Since the sequence 0, 1, 0, 1, . . . does not converge, the infinite series
P
∞
k+1 diverges.
k=1 (−1)
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
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The geometric series
Definition
Let a, r 6= 0. Any series of the form
∞
X
ar k−1 = a + ar 1 + ar 2 + ar 3 + · · ·
k=1
is called a geometric series. The number r is called the common ratio of
the series.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
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The geometric series
Definition
Let a, r 6= 0. Any series of the form
∞
X
ar k−1 = a + ar 1 + ar 2 + ar 3 + · · ·
k=1
is called a geometric series. The number r is called the common ratio of
the series.
Theorem
∞
X
k=1
Mark Woodard (Furman U)

= a
k−1
1−r
ar
diverges
§12.2–Infinite series
if |r | < 1
if |r | ≥ 1
Fall 2010
7 / 16
The geometric series
Problem
Evaluate the series
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P∞
k+2 .
k=1 5(1/3)
§12.2–Infinite series
Fall 2010
8 / 16
The geometric series
Problem
Evaluate the series
P∞
k+2 .
k=1 5(1/3)
Solution
The series
5(1/3)3 + 5(1/3)4 + 5(1/3)5 + · · ·
is geometric with a = 5/27 and r = 1/3. Thus the series converges to
(5/27)
5
= .
1 − (1/3)
18
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
8 / 16
The geometric series
Problem
A certain super ball has the property that it will always return to 60
percent of the maximum height of the previous bounce. If a ball is
dropped from 20 feet, how far will it fall?
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
9 / 16
The geometric series
Problem
A certain super ball has the property that it will always return to 60
percent of the maximum height of the previous bounce. If a ball is
dropped from 20 feet, how far will it fall?
Solution
The total distance is the infinite series
20 + 2(.6)(20) + 2(.6)2 (20) + 2(.6)3 (20) + · · ·
|
{z
}
geometric part
24
= 20 +
1 − .6
= 20 + 60
= 80
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
9 / 16
Telescoping series
Problem
Find the sum of the series
∞
X
k=1
Mark Woodard (Furman U)
k2
1
.
+k
§12.2–Infinite series
Fall 2010
10 / 16
Telescoping series
Problem
Find the sum of the series
∞
X
k=1
k2
1
.
+k
Solution
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
10 / 16
Telescoping series
Problem
Find the sum of the series
∞
X
k=1
k2
1
.
+k
Solution
By a partial fraction decomposition,
Mark Woodard (Furman U)
1
k 2 +k
§12.2–Infinite series
=
1
k
−
1
k+1 .
Fall 2010
10 / 16
Telescoping series
Problem
Find the sum of the series
∞
X
k=1
k2
1
.
+k
Solution
By a partial fraction decomposition,
Thus it can be shown that sn = 1 −
Mark Woodard (Furman U)
1
k 2 +k
1
n+1 .
§12.2–Infinite series
=
1
k
−
1
k+1 .
Fall 2010
10 / 16
Telescoping series
Problem
Find the sum of the series
∞
X
k=1
k2
1
.
+k
Solution
By a partial fraction decomposition,
Thus it can be shown that sn = 1 −
1
k 2 +k
1
n+1 .
=
1
k
−
1
k+1 .
Since sn → 1 as n → ∞, we conclude that the series converges to 1.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
10 / 16
The harmonic series
Problem
Show that the harmonic series
∞
X
1
1 1
= 1 + + + ···
k
2 3
k=1
diverges.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
11 / 16
The harmonic series
Solution
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
12 / 16
The harmonic series
Solution
Consider s8 and observe that
1
1 1
1 1 1 1
s8 = 1 + +
+
+
+ + +
2
3 4
5 6 7 8
1
1
1
≥1+ +2 +4
2
4
8
3
=1+
2
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
12 / 16
The harmonic series
Solution
Consider s8 and observe that
1
1 1
1 1 1 1
s8 = 1 + +
+
+
+ + +
2
3 4
5 6 7 8
1
1
1
≥1+ +2 +4
2
4
8
3
=1+
2
In general, s2n ≥ 1 + n2 , which show that sn → ∞ as n → ∞.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
12 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
Mark Woodard (Furman U)
P∞
n=1 an
diverges.
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
P∞
n=1 an
diverges.
Proof.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
P∞
n=1 an
diverges.
Proof.
We will prove the contrapositive; namely, if the series converges, then
the nth term converges to 0.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
P∞
n=1 an
diverges.
Proof.
We will prove the contrapositive; namely, if the series converges, then
the nth term converges to 0.
P
Let us assume that ∞
k=1 ak converges and let {sn } denote the
corresponding sequence of partial sums.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
P∞
n=1 an
diverges.
Proof.
We will prove the contrapositive; namely, if the series converges, then
the nth term converges to 0.
P
Let us assume that ∞
k=1 ak converges and let {sn } denote the
corresponding sequence of partial sums.
Thus sn → L for some limit L.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
P∞
n=1 an
diverges.
Proof.
We will prove the contrapositive; namely, if the series converges, then
the nth term converges to 0.
P
Let us assume that ∞
k=1 ak converges and let {sn } denote the
corresponding sequence of partial sums.
Thus sn → L for some limit L.
It is also the case that sn−1 → L.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Theorem (nth Term Test for Divergence)
If an 6→ 0, then the series
P∞
n=1 an
diverges.
Proof.
We will prove the contrapositive; namely, if the series converges, then
the nth term converges to 0.
P
Let us assume that ∞
k=1 ak converges and let {sn } denote the
corresponding sequence of partial sums.
Thus sn → L for some limit L.
It is also the case that sn−1 → L.
Thus an = (sn − sn−1 ) → L − L = 0, as was to be shown.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
13 / 16
A test for divergence
Problem
Does the series
P∞
k=1 (1
Mark Woodard (Furman U)
+ k −1 )k converge?
§12.2–Infinite series
Fall 2010
14 / 16
A test for divergence
Problem
Does the series
P∞
k=1 (1
+ k −1 )k converge?
Solution
Since (1 + k −1 )k → e 6= 0, the series diverges by the nth Term Test.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
14 / 16
Some useful theorems on series
Theorem
P
P
Suppose that
an and
bn are convergent series and let c be a
constant. Then
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
15 / 16
Some useful theorems on series
Theorem
P
P
Suppose that
an and
bn are convergent series and let c be a
constant. Then
P
P
P
(an + bn ) = an + bn
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
15 / 16
Some useful theorems on series
Theorem
P
P
Suppose that
an and
bn are convergent series and let c be a
constant. Then
P
P
P
(an + bn ) = an + bn
P
P
P
(an − bn ) = an − bn
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
15 / 16
Some useful theorems on series
Theorem
P
P
Suppose that
an and
bn are convergent series and let c be a
constant. Then
P
P
P
(an + bn ) = an + bn
P
P
P
(an − bn ) = an − bn
P
P
can = c an
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
15 / 16
Some useful theorems on series
Problem
Find the sum of the series
∞ X
k=1
Mark Woodard (Furman U)
1
k+3
+ 5(.5)
.
k2 + k
§12.2–Infinite series
Fall 2010
16 / 16
Some useful theorems on series
Problem
Find the sum of the series
∞ X
k=1
1
k+3
+ 5(.5)
.
k2 + k
Solution
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
16 / 16
Some useful theorems on series
Problem
Find the sum of the series
∞ X
k=1
1
k+3
+ 5(.5)
.
k2 + k
Solution
We have already observed that
Mark Woodard (Furman U)
P∞
1
k=1 k 2 +k
§12.2–Infinite series
= 1.
Fall 2010
16 / 16
Some useful theorems on series
Problem
Find the sum of the series
∞ X
k=1
1
k+3
+ 5(.5)
.
k2 + k
Solution
P
1
We have already observed that ∞
k=1 k 2 +k = 1.
P∞
The series k=1 5(.5)k+3 = 5/8, being geometric.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
16 / 16
Some useful theorems on series
Problem
Find the sum of the series
∞ X
k=1
1
k+3
+ 5(.5)
.
k2 + k
Solution
P
1
We have already observed that ∞
k=1 k 2 +k = 1.
P∞
The series k=1 5(.5)k+3 = 5/8, being geometric.
Thus the original series converges to 1 + 5/8 = 13/8.
Mark Woodard (Furman U)
§12.2–Infinite series
Fall 2010
16 / 16