Calculus Volume 2

# 6.1Power Series and Functions

Calculus Volume 26.1 Power Series and Functions

### Learning Objectives

• 6.1.1. Identify a power series and provide examples of them.
• 6.1.2. Determine the radius of convergence and interval of convergence of a power series.
• 6.1.3. Use a power series to represent a function.

A power series is a type of series with terms involving a variable. More specifically, if the variable is x, then all the terms of the series involve powers of x. As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions. In this section we define power series and show how to determine when a power series converges and when it diverges. We also show how to represent certain functions using power series.

### Form of a Power Series

A series of the form

$∑n=0∞cnxn=c0+c1x+c2x2+⋯,∑n=0∞cnxn=c0+c1x+c2x2+⋯,$

where x is a variable and the coefficients cn are constants, is known as a power series. The series

$1+x+x2+⋯=∑n=0∞xn1+x+x2+⋯=∑n=0∞xn$

is an example of a power series. Since this series is a geometric series with ratio $r=x,r=x,$ we know that it converges if $|x|<1|x|<1$ and diverges if $|x|≥1.|x|≥1.$

### Definition

A series of the form

$∑n=0∞cnxn=c0+c1x+c2x2+⋯∑n=0∞cnxn=c0+c1x+c2x2+⋯$
6.1

is a power series centered at $x=0.x=0.$ A series of the form

$∑n=0∞cn(x−a)n=c0+c1(x−a)+c2(x−a)2+⋯∑n=0∞cn(x−a)n=c0+c1(x−a)+c2(x−a)2+⋯$
6.2

is a power series centered at $x=a.x=a.$

To make this definition precise, we stipulate that $x0=1x0=1$ and $(x−a)0=1(x−a)0=1$ even when $x=0x=0$ and $x=a,x=a,$ respectively.

The series

$∑n=0∞xnn!=1+x+x22!+x33!+⋯∑n=0∞xnn!=1+x+x22!+x33!+⋯$

and

$∑n=0∞n!xn=1+x+2!x2+3!x3+⋯∑n=0∞n!xn=1+x+2!x2+3!x3+⋯$

are both power series centered at $x=0.x=0.$ The series

$∑n=0∞(x−2)n(n+1)3n=1+x−22·3+(x−2)23·32+(x−2)34·33+⋯∑n=0∞(x−2)n(n+1)3n=1+x−22·3+(x−2)23·32+(x−2)34·33+⋯$

is a power series centered at $x=2.x=2.$

### Convergence of a Power Series

Since the terms in a power series involve a variable x, the series may converge for certain values of x and diverge for other values of x. For a power series centered at $x=a,x=a,$ the value of the series at $x=ax=a$ is given by $c0.c0.$ Therefore, a power series always converges at its center. Some power series converge only at that value of x. Most power series, however, converge for more than one value of x. In that case, the power series either converges for all real numbers x or converges for all x in a finite interval. For example, the geometric series $∑n=0∞xn∑n=0∞xn$ converges for all x in the interval $(−1,1),(−1,1),$ but diverges for all x outside that interval. We now summarize these three possibilities for a general power series.

Theorem 6.1

#### Convergence of a Power Series

Consider the power series $∑n=0∞cn(x−a)n.∑n=0∞cn(x−a)n.$ The series satisfies exactly one of the following properties:

1. The series converges at $x=ax=a$ and diverges for all $x≠a.x≠a.$
2. The series converges for all real numbers x.
3. There exists a real number $R>0R>0$ such that the series converges if $|x−a| and diverges if $|x−a|>R.|x−a|>R.$ At the values x where $|x−a|=R,|x−a|=R,$ the series may converge or diverge.

#### Proof

Suppose that the power series is centered at $a=0.a=0.$ (For a series centered at a value of a other than zero, the result follows by letting $y=x−ay=x−a$ and considering the series $∑n=1∞cnyn.)∑n=1∞cnyn.)$ We must first prove the following fact:

If there exists a real number $d≠0d≠0$ such that $∑n=0∞cndn∑n=0∞cndn$ converges, then the series $∑n=0∞cnxn∑n=0∞cnxn$ converges absolutely for all x such that $|x|<|d|.|x|<|d|.$

Since $∑n=0∞cndn∑n=0∞cndn$ converges, the nth term $cndn→0cndn→0$ as $n→∞.n→∞.$ Therefore, there exists an integer N such that $|cndn|≤1|cndn|≤1$ for all $n≥N.n≥N.$ Writing

$|cnxn|=|cndn||xd|n,|cnxn|=|cndn||xd|n,$

we conclude that, for all $n≥N,n≥N,$

$|cnxn|≤|xd|n.|cnxn|≤|xd|n.$

The series

$∑n=N∞|xd|n∑n=N∞|xd|n$

is a geometric series that converges if $|xd|<1.|xd|<1.$ Therefore, by the comparison test, we conclude that $∑n=N∞cnxn∑n=N∞cnxn$ also converges for $|x|<|d|.|x|<|d|.$ Since we can add a finite number of terms to a convergent series, we conclude that $∑n=0∞cnxn∑n=0∞cnxn$ converges for $|x|<|d|.|x|<|d|.$

With this result, we can now prove the theorem. Consider the series

$∑n=0∞anxn∑n=0∞anxn$

and let S be the set of real numbers for which the series converges. Suppose that the set $S={0}.S={0}.$ Then the series falls under case i. Suppose that the set S is the set of all real numbers. Then the series falls under case ii. Suppose that $S≠{0}S≠{0}$ and S is not the set of real numbers. Then there exists a real number $x*≠0x*≠0$ such that the series does not converge. Thus, the series cannot converge for any x such that $|x|>|x*|.|x|>|x*|.$ Therefore, the set S must be a bounded set, which means that it must have a smallest upper bound. (This fact follows from the Least Upper Bound Property for the real numbers, which is beyond the scope of this text and is covered in real analysis courses.) Call that smallest upper bound R. Since $S≠{0},S≠{0},$ the number $R>0.R>0.$ Therefore, the series converges for all x such that $|x| and the series falls into case iii.

If a series $∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n$ falls into case iii. of Convergence of a Power Series, then the series converges for all x such that $|x−a| for some $R>0,R>0,$ and diverges for all x such that $|x−a|>R.|x−a|>R.$ The series may converge or diverge at the values x where $|x−a|=R.|x−a|=R.$ The set of values x for which the series $∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n$ converges is known as the interval of convergence. Since the series diverges for all values x where $|x−a|>R,|x−a|>R,$ the length of the interval is 2R, and therefore, the radius of the interval is R. The value R is called the radius of convergence. For example, since the series $∑n=0∞xn∑n=0∞xn$ converges for all values x in the interval $(−1,1)(−1,1)$ and diverges for all values x such that $|x|≥1,|x|≥1,$ the interval of convergence of this series is $(−1,1).(−1,1).$ Since the length of the interval is 2, the radius of convergence is 1.

### Definition

Consider the power series $∑n=0∞cn(x−a)n.∑n=0∞cn(x−a)n.$ The set of real numbers x where the series converges is the interval of convergence. If there exists a real number $R>0R>0$ such that the series converges for $|x−a| and diverges for $|x−a|>R,|x−a|>R,$ then R is the radius of convergence. If the series converges only at $x=a,x=a,$ we say the radius of convergence is $R=0.R=0.$ If the series converges for all real numbers x, we say the radius of convergence is $R=∞R=∞$ (Figure 6.2).

Figure 6.2 For a series $∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n$ graph (a) shows a radius of convergence at $R=0,R=0,$ graph (b) shows a radius of convergence at $R=∞,R=∞,$ and graph (c) shows a radius of convergence at R. For graph (c) we note that the series may or may not converge at the endpoints $x=a+Rx=a+R$ and $x=a−R.x=a−R.$

To determine the interval of convergence for a power series, we typically apply the ratio test. In Example 6.1, we show the three different possibilities illustrated in Figure 6.2.

### Example 6.1

#### Finding the Interval and Radius of Convergence

For each of the following series, find the interval and radius of convergence.

1. $∑n=0∞xnn!∑n=0∞xnn!$
2. $∑n=0∞n!xn∑n=0∞n!xn$
3. $∑n=0∞(x−2)n(n+1)3n∑n=0∞(x−2)n(n+1)3n$

### Checkpoint 6.1

Find the interval and radius of convergence for the series $∑n=1∞xnn.∑n=1∞xnn.$

### Representing Functions as Power Series

Being able to represent a function by an “infinite polynomial” is a powerful tool. Polynomial functions are the easiest functions to analyze, since they only involve the basic arithmetic operations of addition, subtraction, multiplication, and division. If we can represent a complicated function by an infinite polynomial, we can use the polynomial representation to differentiate or integrate it. In addition, we can use a truncated version of the polynomial expression to approximate values of the function. So, the question is, when can we represent a function by a power series?

Consider again the geometric series

$1+x+x2+x3+⋯=∑n=0∞xn.1+x+x2+x3+⋯=∑n=0∞xn.$
6.3

Recall that the geometric series

$a+ar+ar2+ar3+⋯a+ar+ar2+ar3+⋯$

converges if and only if $|r|<1.|r|<1.$ In that case, it converges to $a1−r.a1−r.$ Therefore, if $|x|<1,|x|<1,$ the series in Example 6.3 converges to $11−x11−x$ and we write

$1+x+x2+x3+⋯=11−xfor|x|<1.1+x+x2+x3+⋯=11−xfor|x|<1.$

As a result, we are able to represent the function $f(x)=11−xf(x)=11−x$ by the power series

$1+x+x2+x3+⋯when|x|<1.1+x+x2+x3+⋯when|x|<1.$

We now show graphically how this series provides a representation for the function $f(x)=11−xf(x)=11−x$ by comparing the graph of f with the graphs of several of the partial sums of this infinite series.

### Example 6.2

#### Graphing a Function and Partial Sums of its Power Series

Sketch a graph of $f(x)=11−xf(x)=11−x$ and the graphs of the corresponding partial sums $SN(x)=∑n=0NxnSN(x)=∑n=0Nxn$ for $N=2,4,6N=2,4,6$ on the interval $(−1,1).(−1,1).$ Comment on the approximation $SNSN$ as N increases.

Checkpoint 6.2

Sketch a graph of $f(x)=11−x2f(x)=11−x2$ and the corresponding partial sums $SN(x)=∑n=0Nx2nSN(x)=∑n=0Nx2n$ for $N=2,4,6N=2,4,6$ on the interval $(−1,1).(−1,1).$

Next we consider functions involving an expression similar to the sum of a geometric series and show how to represent these functions using power series.

### Example 6.3

#### Representing a Function with a Power Series

Use a power series to represent each of the following functions $f.f.$ Find the interval of convergence.

1. $f(x)=11+x3f(x)=11+x3$
2. $f(x)=x24−x2f(x)=x24−x2$
Checkpoint 6.3

Represent the function $f(x)=x32−xf(x)=x32−x$ using a power series and find the interval of convergence.

In the remaining sections of this chapter, we will show ways of deriving power series representations for many other functions, and how we can make use of these representations to evaluate, differentiate, and integrate various functions.

### Section 6.1 Exercises

In the following exercises, state whether each statement is true, or give an example to show that it is false.

1.

If $∑n=1∞anxn∑n=1∞anxn$ converges, then $anxn→0anxn→0$ as $n→∞.n→∞.$

2.

$∑n=1∞anxn∑n=1∞anxn$ converges at $x=0x=0$ for any real numbers $an.an.$

3.

Given any sequence $an,an,$ there is always some $R>0,R>0,$ possibly very small, such that $∑n=1∞anxn∑n=1∞anxn$ converges on $(−R,R).(−R,R).$

4.

If $∑n=1∞anxn∑n=1∞anxn$ has radius of convergence $R>0R>0$ and if $|bn|≤|an||bn|≤|an|$ for all n, then the radius of convergence of $∑n=1∞bnxn∑n=1∞bnxn$ is greater than or equal to R.

5.

Suppose that $∑n=0∞an(x−3)n∑n=0∞an(x−3)n$ converges at $x=6.x=6.$ At which of the following points must the series also converge? Use the fact that if $∑an(x−c)n∑an(x−c)n$ converges at x, then it converges at any point closer to c than x.

1. $x=1x=1$
2. $x=2x=2$
3. $x=3x=3$
4. $x=0x=0$
5. $x=5.99x=5.99$
6. $x=0.000001x=0.000001$
6.

Suppose that $∑n=0∞an(x+1)n∑n=0∞an(x+1)n$ converges at $x=−2.x=−2.$ At which of the following points must the series also converge? Use the fact that if $∑an(x−c)n∑an(x−c)n$ converges at x, then it converges at any point closer to c than x.

1. $x=2x=2$
2. $x=−1x=−1$
3. $x=−3x=−3$
4. $x=0x=0$
5. $x=0.99x=0.99$
6. $x=0.000001x=0.000001$

In the following exercises, suppose that $|an+1an|→1|an+1an|→1$ as $n→∞.n→∞.$ Find the radius of convergence for each series.

7.

$∑n=0∞an2nxn∑n=0∞an2nxn$

8.

$∑n=0∞anxn2n∑n=0∞anxn2n$

9.

$∑n=0∞anπnxnen∑n=0∞anπnxnen$

10.

$∑n=0∞an(−1)nxn10n∑n=0∞an(−1)nxn10n$

11.

$∑n=0∞an(−1)nx2n∑n=0∞an(−1)nx2n$

12.

$∑n=0∞an(−4)nx2n∑n=0∞an(−4)nx2n$

In the following exercises, find the radius of convergence R and interval of convergence for $∑anxn∑anxn$ with the given coefficients $an.an.$

13.

$∑n=1∞(2x)nn∑n=1∞(2x)nn$

14.

$∑n=1∞(−1)nxnn∑n=1∞(−1)nxnn$

15.

$∑n=1∞nxn2n∑n=1∞nxn2n$

16.

$∑n=1∞nxnen∑n=1∞nxnen$

17.

$∑n=1∞n2xn2n∑n=1∞n2xn2n$

18.

$∑k=1∞kexkek∑k=1∞kexkek$

19.

$∑k=1∞πkxkkπ∑k=1∞πkxkkπ$

20.

$∑n=1∞xnn!∑n=1∞xnn!$

21.

$∑n=1∞10nxnn!∑n=1∞10nxnn!$

22.

$∑n=1∞(−1)nxnln(2n)∑n=1∞(−1)nxnln(2n)$

In the following exercises, find the radius of convergence of each series.

23.

$∑k=1∞(k!)2xk(2k)!∑k=1∞(k!)2xk(2k)!$

24.

$∑n=1∞(2n)!xnn2n∑n=1∞(2n)!xnn2n$

25.

$∑k=1∞k!1·3·5⋯(2k−1)xk∑k=1∞k!1·3·5⋯(2k−1)xk$

26.

$∑k=1∞2·4·6⋯2k(2k)!xk∑k=1∞2·4·6⋯2k(2k)!xk$

27.

$∑n=1∞xn(2nn)∑n=1∞xn(2nn)$ where $(nk)=n!k!(n−k)!(nk)=n!k!(n−k)!$

28.

$∑n=1∞sin2nxn∑n=1∞sin2nxn$

In the following exercises, use the ratio test to determine the radius of convergence of each series.

29.

$∑n=1∞(n!)3(3n)!xn∑n=1∞(n!)3(3n)!xn$

30.

$∑n=1∞23n(n!)3(3n)!xn∑n=1∞23n(n!)3(3n)!xn$

31.

$∑n=1∞n!nnxn∑n=1∞n!nnxn$

32.

$∑n=1∞(2n)!n2nxn∑n=1∞(2n)!n2nxn$

In the following exercises, given that $11−x=∑n=0∞xn11−x=∑n=0∞xn$ with convergence in $(−1,1),(−1,1),$ find the power series for each function with the given center a, and identify its interval of convergence.

33.

$f(x)=1x;a=1f(x)=1x;a=1$ (Hint: $1x=11−(1−x))1x=11−(1−x))$

34.

$f(x)=11−x2;a=0f(x)=11−x2;a=0$

35.

$f(x)=x1−x2;a=0f(x)=x1−x2;a=0$

36.

$f(x)=11+x2;a=0f(x)=11+x2;a=0$

37.

$f(x)=x21+x2;a=0f(x)=x21+x2;a=0$

38.

$f(x)=12−x;a=1f(x)=12−x;a=1$

39.

$f(x)=11−2x;a=0.f(x)=11−2x;a=0.$

40.

$f(x)=11−4x2;a=0f(x)=11−4x2;a=0$

41.

$f(x)=x21−4x2;a=0f(x)=x21−4x2;a=0$

42.

$f(x)=x25−4x+x2;a=2f(x)=x25−4x+x2;a=2$

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.

43.

Explain why, if $|an|1/n→r>0,|an|1/n→r>0,$ then $|anxn|1/n→|x|r<1|anxn|1/n→|x|r<1$ whenever $|x|<1r|x|<1r$ and, therefore, the radius of convergence of $∑n=1∞anxn∑n=1∞anxn$ is $R=1r.R=1r.$

44.

$∑n=1∞xnnn∑n=1∞xnnn$

45.

$∑k=1∞(k−12k+3)kxk∑k=1∞(k−12k+3)kxk$

46.

$∑k=1∞(2k2−1k2+3)kxk∑k=1∞(2k2−1k2+3)kxk$

47.

$∑n=1∞an=(n1/n−1)nxn∑n=1∞an=(n1/n−1)nxn$

48.

Suppose that $p(x)=∑n=0∞anxnp(x)=∑n=0∞anxn$ such that $an=0an=0$ if n is odd. Explain why $p(x)=−p(−x).p(x)=−p(−x).$

49.

Suppose that $p(x)=∑n=0∞anxnp(x)=∑n=0∞anxn$ such that $an=0an=0$ if n is even. Explain why $p(x)=p(−x).p(x)=p(−x).$

50.

Suppose that $p(x)=∑n=0∞anxnp(x)=∑n=0∞anxn$ converges on $(−1,1].(−1,1].$ Find the interval of convergence of $p(Ax).p(Ax).$

51.

Suppose that $p(x)=∑n=0∞anxnp(x)=∑n=0∞anxn$ converges on $(−1,1].(−1,1].$ Find the interval of convergence of $p(2x−1).p(2x−1).$

In the following exercises, suppose that $p(x)=∑n=0∞anxnp(x)=∑n=0∞anxn$ satisfies $limn→∞an+1an=1limn→∞an+1an=1$ where $an≥0an≥0$ for each n. State whether each series converges on the full interval $(−1,1),(−1,1),$ or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.

52.

$∑n=0∞anx2n∑n=0∞anx2n$

53.

$∑n=0∞a2nx2n∑n=0∞a2nx2n$

54.

$∑n=0∞a2nxn(Hint:x=±x2)∑n=0∞a2nxn(Hint:x=±x2)$

55.

$∑n=0∞an2xn2∑n=0∞an2xn2$ (Hint: Let $bk=akbk=ak$ if $k=n2k=n2$ for some n, otherwise $bk=0.)bk=0.)$

56.

Suppose that $p(x)p(x)$ is a polynomial of degree N. Find the radius and interval of convergence of $∑n=1∞p(n)xn.∑n=1∞p(n)xn.$

57.

[T] Plot the graphs of $11−x11−x$ and of the partial sums $SN=∑n=0NxnSN=∑n=0Nxn$ for $n=10,20,30n=10,20,30$ on the interval $[−0.99,0.99].[−0.99,0.99].$ Comment on the approximation of $11−x11−x$ by $SNSN$ near $x=−1x=−1$ and near $x=1x=1$ as N increases.

58.

[T] Plot the graphs of $−ln(1−x)−ln(1−x)$ and of the partial sums $SN=∑n=1NxnnSN=∑n=1Nxnn$ for $n=10,50,100n=10,50,100$ on the interval $[−0.99,0.99].[−0.99,0.99].$ Comment on the behavior of the sums near $x=−1x=−1$ and near $x=1x=1$ as N increases.

59.

[T] Plot the graphs of the partial sums $Sn=∑n=1Nxnn2Sn=∑n=1Nxnn2$ for $n=10,50,100n=10,50,100$ on the interval $[−0.99,0.99].[−0.99,0.99].$ Comment on the behavior of the sums near $x=−1x=−1$ and near $x=1x=1$ as N increases.

60.

[T] Plot the graphs of the partial sums $SN=∑n=1NsinnxnSN=∑n=1Nsinnxn$ for $n=10,50,100n=10,50,100$ on the interval $[−0.99,0.99].[−0.99,0.99].$ Comment on the behavior of the sums near $x=−1x=−1$ and near $x=1x=1$ as N increases.

61.

[T] Plot the graphs of the partial sums $SN=∑n=0N(−1)nx2n+1(2n+1)!SN=∑n=0N(−1)nx2n+1(2n+1)!$ for $n=3,5,10n=3,5,10$ on the interval $[−2π,2π].[−2π,2π].$ Comment on how these plots approximate $sinxsinx$ as N increases.

62.

[T] Plot the graphs of the partial sums $SN=∑n=0N(−1)nx2n(2n)!SN=∑n=0N(−1)nx2n(2n)!$ for $n=3,5,10n=3,5,10$ on the interval $[−2π,2π].[−2π,2π].$ Comment on how these plots approximate $cosxcosx$ as N increases.