Calculus Volume 2

Gilbert Strang,

Chapter 6

Power Series - all with Video Answers

Educators

+ 2 more educators

Section 1

Power Series and Functions

Problem 1

In the following exercises, state whether each statement is true, or give an example to show that it is false.
If $\sum_{n=1}^{\infty} a_{n} x^{n}$ converges, then $a_{n} x^{n} \rightarrow 0$ as $n \rightarrow \infty$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

In the following exercises, state whether each statement is true, or give an example to show that it is false.
$\sum_{n=1}^{\infty} a_{n} x^{n}$ converges at $x=0$ for any real numbers

Joshua Utley

Numerade Educator

Problem 3

In the following exercises, state whether each statement is true, or give an example to show that it is false.
Given any sequence $a_{n},$ there is always some $R > 0$ possibly very small, such that $\sum_{n=1}^{\infty} a_{n} x^{n}$ converges on $(-R, R) .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

In the following exercises, state whether each statement is true, or give an example to show that it is false.
If $\sum_{n=1}^{\infty} a_{n} x^{n}$ has radius of convergence $R > 0$ and if $\left|b_{n}\right| \leq\left|a_{n}\right|$ for all $n,$ then the radius of convergence of $\sum_{n=1}^{\infty} b_{n} x^{n}$ is greater than or equal to $R$

Joshua Utley

Numerade Educator

Problem 5

Suppose that $\sum_{n=0}^{\infty} a_{n}(x-3)^{n}$ converges at $x=6 .$ At which of the following points must the series also converge? Use the fact that if $\sum a_{n}(x-c)^{n}$ converges at
$x,$ then it converges at any point closer to $c$ than $x$ .
a. $x=1$
b. $x=2$
c. $x=3$
d. $x=0$
e. $x=5.99$
f. $x=0.000001$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

Suppose that $\sum_{n=0}^{\infty} a_{n}(x+1)^{n}$ converges at $x=-2$ At which of the following points must the series also converge? Use the fact that if $\sum a_{n}(x-c)^{n}$ converges at $x,$ then it converges at any point closer to $c$ than $x$ .
a. x = 2
b. x = ?1
c. x = ?3
d. x = 0
e. x = 0.99
f. x = 0.000001

Joshua Utley

Numerade Educator

Problem 7

In the following exercises, suppose that $\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1$ as
$n \rightarrow \infty .$ Find the radius of convergence for each series.
$$\sum_{n=0}^{\infty} a_{n} 2^{n} x^{n}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

In the following exercises, suppose that $\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1$ as
$n \rightarrow \infty .$ Find the radius of convergence for each series.
$$\sum_{n=0}^{\infty} \frac{a_{n} x^{n}}{2^{n}}$$

Joshua Utley

Numerade Educator

Problem 9

In the following exercises, suppose that $\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1$ as
$n \rightarrow \infty .$ Find the radius of convergence for each series.
$$\sum_{n=0}^{\infty} \frac{a_{n} \pi^{n} x^{n}}{e^{n}}$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 10

In the following exercises, suppose that $\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1$ as
$n \rightarrow \infty .$ Find the radius of convergence for each series.
$$\sum_{n=0}^{\infty} \frac{a_{n}(-1)^{n} x^{n}}{10^{n}}$$

Joshua Utley

Numerade Educator

Problem 11

In the following exercises, suppose that $\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1$ as
$n \rightarrow \infty .$ Find the radius of convergence for each series.
$$\sum_{n=0}^{\infty} a_{n}(-1)^{n} x^{2 n}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

In the following exercises, suppose that $\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1$ as
$n \rightarrow \infty .$ Find the radius of convergence for each series.
$$\sum_{n=0}^{\infty} a_{n}(-4)^{n} x^{2 n}$$

Joshua Utley

Numerade Educator

Problem 13

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty} \frac{(2 x)^{n}}{n}$$

Vincenzo Zaccaro

Vincenzo Zaccaro

Numerade Educator

Problem 14

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty}(-1)^{n} \frac{x^{n}}{\sqrt{n}}$$

Joshua Utley

Numerade Educator

Problem 15

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty} \frac{n x^{n}}{2^{n}}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty} \frac{n x^{n}}{e^{n}}$$

Joshua Utley

Numerade Educator

Problem 17

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty} \frac{n^{2} x^{n}}{2^{n}}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{k=1}^{\infty} \frac{k^{e} x^{k}}{e^{k}}$$

Joshua Utley

Numerade Educator

Problem 19

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{k=1}^{\infty} \frac{\pi^{k} x^{k}}{k^{\pi}}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty} \frac{x^{n}}{n !}$$

Joshua Utley

Numerade Educator

Problem 21

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty} \frac{10^{n} x^{n}}{n !}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

In the following exercises, find the radius of convergence $R$ and interval of convergence for $\sum a_{n} x^{n}$ with the given coefficients $a_{n}$
$$\sum_{n=1}^{\infty}(-1)^{n} \frac{x^{n}}{n(2 n)}$$

Joshua Utley

Numerade Educator

Problem 23

In the following exercises, find the radius of convergence of each series,
$$\sum_{k=1}^{\infty} \frac{(k !)^{2} x^{k}}{(2 k) !}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

In the following exercises, find the radius of convergence of each series,
$$\sum_{n=1}^{\infty} \frac{(2 n) ! x^{n}}{n^{2 n}}$$

Joshua Utley

Numerade Educator

Problem 25

In the following exercises, find the radius of convergence of each series,
$$\sum_{k=1}^{\infty} \frac{k !}{1 \cdot 3 \cdot 5 \cdots(2 k-1)} x^{k}$$

Norman Atentar

Numerade Educator

Problem 26

In the following exercises, find the radius of convergence of each series,
$$\sum_{k=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot .2 k_{x}}{(2 k) !} x^{k}$$

Joshua Utley

Numerade Educator

Problem 27

In the following exercises, find the radius of convergence of each series,
$$\sum_{n=1}^{\infty} \frac{x^{n}}{(2 n)} \text { where }\left(\begin{array}{l}{n} \\ {k}\end{array}\right)=\frac{n !}{k !(n-k) !}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

In the following exercises, find the radius of convergence of each series,
$$\sum_{n=1}^{\infty} \sin ^{2} n x^{n}$$

Joshua Utley

Numerade Educator

Problem 29

In the following exercises, use the ratio test to determine the radius of convergence of each series.
$$\sum_{n=1}^{\infty} \frac{(n !)^{3}}{(3 n) !} x^{n}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

In the following exercises, use the ratio test to determine the radius of convergence of each series.
$$\sum_{n=1}^{\infty} \frac{2^{3 n}(n !)^{3}}{(3 n) !} x^{n}$$

Joshua Utley

Numerade Educator

Problem 31

In the following exercises, use the ratio test to determine the radius of convergence of each series.
$$\sum_{n=1}^{\infty} \frac{n !}{n^{n}} x^{n}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

In the following exercises, use the ratio test to determine the radius of convergence of each series.
$$\sum_{n=1}^{\infty} \frac{(2 n) !}{n^{2 n}} x^{n}$$

Joshua Utley

Numerade Educator

Problem 33

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{1}{x} ; a=1\left(\text { Hint: } \frac{1}{x}=\frac{1}{1-(1-x)}\right) $$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{1}{1-x^{2}} ; a=0$$

Joshua Utley

Numerade Educator

Problem 35

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{x}{1-x^{2}} ; a=0$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{1}{1+x^{2}} ; a=0$$

Joshua Utley

Numerade Educator

Problem 37

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{x^{2}}{1+x^{2}} ; a=0$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{1}{2-x} ; a=1$$

Joshua Utley

Numerade Educator

Problem 39

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{1}{1-2 x} ; a=0$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{1}{1-4 x^{2}} ; a=0$$

Joshua Utley

Numerade Educator

Problem 41

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{x^{2}}{1-4 x^{2}} ; a=0$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

In the following exercises, given that $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ with convergence in $(-1,1),$ find the power series for each function with the given center $a,$ and identify its interval of convergence.
$$f(x)=\frac{x^{2}}{5-4 x+x^{2}} ; a=2$$

Joshua Utley

Numerade Educator

Problem 43

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.
Explain why, if $\quad\left|a_{n}\right|^{1 / n} \rightarrow r > 0, \quad$ then $\left|a_{n} x^{n}\right|^{1 / n} \rightarrow|x| r < 1$ whenever $|x| < \frac{1}{r}$ and, therefore, the radius of convergence of $\sum_{n=1} a_{n} x^{n}$ is $R=\frac{1}{r}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.
$$\sum_{n=1}^{\infty} \frac{x^{n}}{n^{n}}$$

Joshua Utley

Numerade Educator

Problem 45

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.
$$\sum_{k=1}^{\infty}\left(\frac{k-1}{2 k+3}\right)^{k} x^{k}$$

SH

Shawna Haider

Numerade Educator

Problem 46

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.
$$\sum_{k=1}^{\infty}\left(\frac{2 k^{2}-1}{k^{2}+3}\right)^{k} x^{k}$$

Joshua Utley

Numerade Educator

Problem 47

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.
$$\sum_{n=1}^{\infty} a_{n}=\left(n^{1 / n}-1\right)^{n} x^{n}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

Suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ such that $a_{n}=0$ if $n$ is even, Explain why $p(x)=p(-x)$

Joshua Utley

Numerade Educator

Problem 49

Suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ such that $a_{n}=0$ if $n$ is odd. Explain why $p(x)=-p(-x)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

Suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n} \quad$ converges on $(-1,1] .$ Find the interval of convergence of $p(A x) .$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 51

Suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n} \quad$ converges on $(-1,1] .$ Find the interval of convergence of $p(2 x-1)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

In the following exercises, suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ satisfies $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=1$ where $a_{n} \geq 0$ for each $n .$ State whether each series converges on the full interval $(-1,1),$ or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.
$$\sum_{n=0}^{\infty} a_{n} x^{2 n}$$

Joshua Utley

Numerade Educator

Problem 53

In the following exercises, suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ satisfies $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=1$ where $a_{n} \geq 0$ for each $n .$ State whether each series converges on the full interval $(-1,1),$ or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.
$$\sum_{n=0}^{\infty} a_{2 n} x^{2 n}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

In the following exercises, suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ satisfies $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=1$ where $a_{n} \geq 0$ for each $n .$ State whether each series converges on the full interval $(-1,1),$ or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.
$$\sum_{n=0}^{\infty} a_{2 n} x^{n}\left(\text { Hint: } x=\pm \sqrt{x^{2}}\right)$$

Joshua Utley

Numerade Educator

Problem 55

In the following exercises, suppose that $p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ satisfies $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=1$ where $a_{n} \geq 0$ for each $n .$ State whether each series converges on the full interval $(-1,1),$ or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.
$$\sum_{n=0}^{\infty} a_{n^{2}} x^{n^{2}}\left(\text { Hint: Let } b_{k}=a_{k} \text { if } k=n^{2} \text { for some }\right. n, \text { otherwise } b_{k}=0 . ) $$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

Suppose that p(x) is a polynomial of degree N. Find the radius and interval of convergence of $\sum_{n=1}^{\infty} p(n) x^{n}$

Joshua Utley

Numerade Educator

Problem 57

[T] Plot the graphs of $\frac{1}{1-x}$ and of the partial sums $S_{N}=\sum_{n=0}^{N} x^{n}$ for $n=10,20,30$ on the interval $[-0.99,0.99] .$ Comment on the approximation of $\frac{1}{1-x}$ by $S_{N}$ near $x=-1$ and near $x=1$ as $N$ increases.

Joshua Utley

Numerade Educator

Problem 58

[T] Plot the graphs of $-\ln (1-x)$ and of the partial sums $S_{N}=\sum_{n=1}^{N} \frac{x^{n}}{n}$ for $n=10,50,100$ on the interval $[-0.99,0.99] .$ Comment on the behavior of the sums near $x=-1$ and near $x=1$ as $N$ increases.

Joshua Utley

Numerade Educator

Problem 59

[T] Plot the graphs of the partial sums $S_{n}=\sum_{n=1}^{N} \frac{x^{n}}{n^{2}}$ for n = 10, 50, 100 on the interval [?0.99, 0.99]. Comment on the behavior of the sums near x = ?1 and near x = 1 as N increases.

Joshua Utley

Numerade Educator

Problem 60

[T] Plot the graphs of the partial sums $S_{N}=\sum_{n=1}^{N} \sin n x^{n}$ for $n=10,50,100$ on the interval $[-0.99,0.99] .$ Comment on the behavior of the sums near $x=-1$ and near $x=1$ as $N$ increases.

Joshua Utley

Numerade Educator

Problem 61

[T] Plot the graphs of the partial sums $S_{N}=\sum_{n=0}^{N}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !} \quad$ for $\quad n=3,5,10$ on the interval $[-2 \pi, 2 \pi]$ . Comment on how these plots approximate $\sin x$ as $N$ increases.

Joshua Utley

Numerade Educator

Problem 62

[T] Plot the graphs of the partial sums $S_{N}=\sum_{n=0}^{N}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$ for $n=3,5,10$ on the interval $[-2 \pi, 2 \pi]$ . Comment on how these plots approximate $\cos x$ as $N$ increases.

Joshua Utley

Numerade Educator