$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 1.1E: Exercises

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Exercise $$\PageIndex{1}$$

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

1. If $$\displaystyle \sum_{n=1}^∞a_nx^n$$ converges, then $$\displaystyle a_nx^n→0$$ as $$\displaystyle n→∞.$$

True. If a series converges then its terms tend to zero.

2. $$\displaystyle \sum_{n=1}^∞a_nx^n$$ converges at $$\displaystyle x=0$$ for any real numbers $$\displaystyle a_n$$.

3. Given any sequence $$\displaystyle a_n$$, there is always some $$\displaystyle R>0$$, possibly very small, such that $$\displaystyle \sum_{n=1}^∞a_nx^n$$ converges on $$\displaystyle (−R,R)$$.

False. It would imply that $$\displaystyle a_nx^n→0$$ for $$\displaystyle |x|<R$$. If $$\displaystyle a_n=n^n$$, then $$\displaystyle a_nx^n=(nx)^n$$ does not tend to zero for any $$\displaystyle x≠0$$.

4. If $$\displaystyle \sum_{n=1}^∞a_nx^n$$ has radius of convergence $$\displaystyle R>0$$ and if $$\displaystyle |b_n|≤|a_n|$$ for all $$\displaystyle n$$, then the radius of convergence of $$\displaystyle \sum_{n=1}^∞b_nx^n$$ is greater than or equal to $$\displaystyle R$$

## Exercise $$\PageIndex{2}$$

1. Suppose that $$\displaystyle \sum_{n=0}^∞a_n(x−3)^n$$ converges at $$\displaystyle x=6$$. At which of the following points must the series also converge? Use the fact that if $$\displaystyle \sum a_n(x−c)^n$$ converges at $$\displaystyle x$$, then it converges at any point closer to $$\displaystyle c$$ than $$\displaystyle x$$.

a. $$\displaystyle x=1$$

b. $$\displaystyle x=2$$

c. $$\displaystyle x=3$$

d. $$\displaystyle x=0$$

e. $$\displaystyle x=5.99$$

f. $$\displaystyle x=0.000001$$

It must converge on $$\displaystyle (0,6]$$ and hence at: a. $$\displaystyle x=1$$; b. $$\displaystyle x=2$$; c. $$\displaystyle x=3$$; d. $$\displaystyle x=0$$; e. $$\displaystyle x=5.99$$; and f. $$\displaystyle x=0.000001$$.

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

a. $$\displaystyle x=2$$

b. $$\displaystyle x=−1$$

c. $$\displaystyle x=−3$$

d. $$\displaystyle x=0$$

e. $$\displaystyle x=0.99$$

f. $$\displaystyle x=0.000001$$

## Exercise $$\PageIndex{3}$$

In the following exercises, suppose that $$\displaystyle ∣\frac{a_{n+1}}{a_n}∣→1$$ as $$\displaystyle n→∞.$$ Find the radius of convergence for each series.

1. $$\displaystyle \sum_{n=0}^∞a_n2^nx^n$$

$$\displaystyle ∣\frac{a_{n+1}2^{n+1}x^{n+1}}{a_n2^nx^n}∣ =2|x|∣ \frac{a_{n+1}}{a_n}∣→2|x|$$ so $$\displaystyle R=\frac{1}{2}$$

2. $$\displaystyle \sum_{n=0}^∞\frac{a_nx^n}{2^n}$$

$$\displaystyle|\frac{a_{n+1}x^{n+1}}{2^{n+1}} \frac{2^n}{a_nx^n}| = \frac{|x|}{2}| \frac{a_{n+1}}{a_n}∣→\frac{|x|}{2}$$ so $$R=2$$

3. $$\displaystyle \sum_{n=0}^∞\frac{a_nπ^nx^n}{e^n}$$

$$\displaystyle ∣\frac{a_{n+1}(\frac{π}{e})^{n+1}x^{n+1}}{a_n(\frac{π}{e})^nx^n}∣ =\frac{π|x|}{e}∣\frac{a_{n+1}}{a_n}∣→\frac{π|x|}{e}$$ so $$\displaystyle R=\frac{e}{π}$$

4. $$\displaystyle \sum_{n=0}^∞\frac{a_n(−1)^nx^n}{10^n}$$

$$\displaystyle|\frac{a_{n+1}(-1)^{n+1}x^{n+1}}{10^{n+1}} \frac{10^n}{a_n(-1)^nx^n}| = \frac{|x|}{10}| \frac{a_{n+1}}{a_n}∣→\frac{|x|}{10}$$ so $$R=10$$

5. $$\displaystyle \sum_{n=0}^∞a_n(−1)^nx^{2n}$$

$$\displaystyle ∣\frac{a_{n+1}(−1)^{n+1}x^{2n+2}}{a_n(−1)^nx^{2n}}∣ =∣x^2∣∣\frac{a_{n+1}}{a_n}∣→∣x^2∣$$ so $$\displaystyle R=1$$

6. $$\displaystyle \sum_{n=0}^∞a_n(−4)^nx^{2n}$$

$$\displaystyle ∣\frac{a_{n+1}(-4)^{n+1}x^{2(n+1)}}{a_n(-4)^nx^{2n}}∣ =4|x^2|∣ \frac{a_{n+1}}{a_n}∣→4|x^2|$$ so $$\displaystyle R=\frac{1}{2}$$

## Exercise $$\PageIndex{4}$$

In the following exercises, find the radius of convergence $$\displaystyle R$$ and interval of convergence for $$\displaystyle \sum a_nx^n$$ with the given coefficients $$\displaystyle a_n$$.

1. $$\displaystyle \sum_{n=1}^∞\frac{(2x)^n}{n}$$

$$\displaystyle a_n=\frac{2^n}{n}$$ so $$\displaystyle \frac{a_{n+1}x}{a_n}→2x$$. so $$\displaystyle R=\frac{1}{2}$$. When $$\displaystyle x=\frac{1}{2}$$ the series is harmonic and diverges. When $$\displaystyle x=−\frac{1}{2}$$ the series is alternating harmonic and converges. The interval of convergence is $$\displaystyle I=[−\frac{1}{2},\frac{1}{2})$$.

2. $$\displaystyle \sum_{n=1}^∞(−1)^n\frac{x^n}{\sqrt{n}}$$

R=1

Interval of convergence (-1,1)

3. $$\displaystyle \sum_{n=1}^∞\frac{nx^n}{2^n}$$

$$\displaystyle a_n=\frac{n}{2^n}$$ so $$\displaystyle \frac{a_{n+1}x}{a_n}→\frac{x}{2}$$ so $$\displaystyle R=2$$. When $$\displaystyle x=±2$$ the series diverges by the divergence test. The interval of convergence is $$\displaystyle I=(−2,2)$$.

4. $$\displaystyle \sum_{n=1}^∞\frac{nx^n}{e^n}$$

5. $$\displaystyle \sum_{n=1}^∞\frac{n^2x^n}{2^n}$$

$$\displaystyle a_n=\frac{n^2}{2^n}$$ so $$\displaystyle R=2$$. When $$\displaystyle x=±2$$ the series diverges by the divergence test. The interval of convergence is $$\displaystyle I=(−2,2).$$

6. $$\displaystyle \sum_{k=1}^∞\frac{k^ex^k}{e^k}$$

7. $$\displaystyle \sum_{k=1}^∞\frac{π^kx^k}{k^π}$$

$$\displaystyle a_k=\frac{π^k}{k^π}$$ so $$\displaystyle R=\frac{1}{π}$$. When $$\displaystyle x=±\frac{1}{π}$$ the series is an absolutely convergent p-series. The interval of convergence is $$\displaystyle I=[−\frac{1}{π},\frac{1}{π}].$$

8. $$\displaystyle \sum_{n=1}^∞\frac{x^n}{n!}$$

9. $$\displaystyle \sum_{n=1}^∞\frac{10^nx^n}{n!}$$

$$\displaystyle a_n=\frac{10^n}{n!},\frac{a_{n+1}x}{a_n}=\frac{10x}{n+1}→0<1$$ so the series converges for all $$\displaystyle x$$ by the ratio test and $$\displaystyle I=(−∞,∞)$$.

10. $$\displaystyle \sum_{n=1}^∞(−1)^n\frac{x^n}{ln(2n)}$$

## Exercise $$\PageIndex{5}$$

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

1. $$\displaystyle \sum_{k=1}^∞\frac{(k!)^2x^k}{(2k)!}$$

$$\displaystyle a_k=\frac{(k!)^2}{(2k)!}$$ so $$\displaystyle \frac{a_{k+1}}{a_k}=\frac{(k+1)^2}{(2k+2)(2k+1)}→\frac{1}{4}$$ so $$\displaystyle R=4$$

2. $$\displaystyle \sum_{n=1}^∞\frac{(2n)!x^n}{n^{2n}}$$

3. $$\displaystyle \sum_{k=1}^∞\frac{k!}{1⋅3⋅5⋯(2k−1)}x^k$$

$$\displaystyle a_k=\frac{k!}{1⋅3⋅5⋯(2k−1)}$$ so $$\displaystyle \frac{a_{k+1}}{a_k}=\frac{k+1}{2k+1}→\frac{1}{2}$$ so $$\displaystyle R=2$$

4. $$\displaystyle \sum_{k=1}^∞\frac{2⋅4⋅6⋯2k}{(2k)!}x^k$$

5. $$\displaystyle \sum_{n=1}^∞\frac{x^n}{(^{2n}_n)}$$ where $$\displaystyle (^n_k)=\frac{n!}{k!(n−k)!}$$

$$\displaystyle a_n=\frac{1}{(^{2n}_n)}$$ so $$\displaystyle \frac{a_{n+1}}{a_n}=\frac{((n+1)!)^2}{(2n+2)!}\frac{2n!}{(n!)^2}=\frac{(n+1)^2}{(2n+2)(2n+1)}→\frac{1}{4}$$ so $$\displaystyle R=4$$

6. $$\displaystyle \sum_{n=1}^∞sin^2nx^n$$

## Exercise $$\PageIndex{6}$$

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

1. $$\displaystyle \sum_{n=1}^∞\frac{(n!)^3}{(3n)!}x^n$$

$$\displaystyle \frac{a_{n+1}}{a_n}=\frac{(n+1)^3}{(3n+3)(3n+2)(3n+1)}→\frac{1}{27}$$ so $$\displaystyle R=27$$

2. $$\displaystyle \sum_{n=1}^∞\frac{2^{3n}(n!)^3}{(3n)!}x^n$$

3. $$\displaystyle \sum_{n=1}^∞\frac{n!}{n^n}x^n$$

$$\displaystyle a_n=\frac{n!}{n^n}$$ so $$\displaystyle \frac{a_{n+1}}{a_n}=\frac{(n+1)!}{n!}\frac{n^n}{(n+1)^{n+1}}=(\frac{n}{n+1})^n→\frac{1}{e}$$ so $$\displaystyle R=e$$

4. $$\displaystyle \sum_{n=1}^∞\frac{(2n)!}{n^{2n}}x^n$$

## Exercise $$\PageIndex{7}$$

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

1. $$\displaystyle f(x)=\frac{1}{x};a=1$$ (Hint: $$\displaystyle \frac{1}{x}=\frac{1}{1−(1−x)})$$

$$\displaystyle f(x)=\sum_{n=0}^∞(1−x)^n$$ on $$\displaystyle I=(0,2)$$

2. $$\displaystyle f(x)=\frac{1}{1−x^2};a=0$$

3. $$\displaystyle f(x)=\frac{x}{1−x^2};a=0$$

$$\displaystyle \sum_{n=0}^∞x^{2n+1}$$ on $$\displaystyle I=(−1,1)$$

4. $$\displaystyle f(x)=\frac{1}{1+x^2};a=0$$

5. $$\displaystyle f(x)=\frac{x^2}{1+x^2};a=0$$

$$\displaystyle \sum_{n=0}^∞(−1)^nx^{2n+2}$$ on $$\displaystyle I=(−1,1)$$

6. $$\displaystyle f(x)=\frac{1}{2−x};a=1$$

7. $$\displaystyle f(x)=\frac{1}{1−2x};a=0.$$

$$\displaystyle \sum_{n=0}^∞2^nx^n$$ on $$\displaystyle (−\frac{1}{2},\frac{1}{2})$$

8. $$\displaystyle f(x)=\frac{1}{1−4x^2};a=0$$

9. $$\displaystyle f(x)=\frac{x^2}{1−4x^2};a=0$$

$$\displaystyle \sum_{n=0}^∞4^nx^{2n+2}$$ on $$\displaystyle (−\frac{1}{2},\frac{1}{2})$$

10. $$\displaystyle f(x)=\frac{x^2}{5−4x+x^2};a=2$$

## Exercise $$\PageIndex{8}$$

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

1. Explain why, if $$\displaystyle |a_n|^{1/n}→r>0,$$ then $$\displaystyle |a_nx^n|^{1/n}→|x|r<1$$ whenever $$\displaystyle |x|<\frac{1}{r}$$ and, therefore, the radius of convergence of $$\displaystyle \sum_{n=1}^∞a_nx^n$$ is $$\displaystyle R=\frac{1}{r}$$.

$$\displaystyle |a_nx^n|^{1/n}=|a_n|^{1/n}|x|→|x|r$$ as $$\displaystyle n→∞$$ and $$\displaystyle |x|r<1$$ when $$\displaystyle |x|<\frac{1}{r}$$. Therefore, $$\displaystyle \sum_{n=1}^∞a_nx^n$$ converges when $$\displaystyle |x|<\frac{1}{r}$$ by the nth root test.

2. $$\displaystyle \sum_{n=1}^∞\frac{x^n}{n^n}$$

3. $$\displaystyle \sum_{k=1}^∞(\frac{k−1}{2k+3})^kx^k$$

$$\displaystyle a_k=(\frac{k−1}{2k+3})^k$$ so $$\displaystyle (a_k)^{1/k}→\frac{1}{2}<1$$ so $$\displaystyle R=2$$

4. $$\displaystyle \sum_{k=1}^∞(\frac{2k^2−1}{k^2+3})^kx^k$$

5. $$\displaystyle \sum_{n=1}^∞a_n=(n^{1/n}−1)^nx^n$$

$$\displaystyle a_n=(n^{1/n}−1)^n$$ so $$\displaystyle (a_n)^{1/n}→0$$ so $$\displaystyle R=∞$$

6. Suppose that $$\displaystyle p(x)=\sum_{n=0}^∞a_nx^n$$ such that $$\displaystyle a_n=0$$ if $$\displaystyle n$$ is even. Explain why $$\displaystyle p(x)=p(−x).$$

7. Suppose that $$\displaystyle p(x)=\sum_{n=0}^∞a_nx^n$$ such that $$\displaystyle a_n=0$$ if $$\displaystyle n$$ is odd. Explain why $$\displaystyle p(x)=−p(−x).$$

We can rewrite $$\displaystyle p(x)=\sum_{n=0}^∞a_{2n+1}x^{2n+1}$$ and $$\displaystyle p(x)=p(−x)$$ since $$\displaystyle x^{2n+1}=−(−x)^{2n+1}$$.

8. Suppose that $$\displaystyle p(x)=\sum_{n=0}^∞a_nx^n$$ converges on $$\displaystyle (−1,1]$$. Find the interval of convergence of $$\displaystyle p(Ax)$$.

9. Suppose that $$\displaystyle p(x)=\sum_{n=0}^∞a_nx^n$$ converges on $$\displaystyle (−1,1]$$. Find the interval of convergence of $$\displaystyle p(2x−1)$$.

If $$\displaystyle x∈[0,1],$$ then $$\displaystyle y=2x−1∈[−1,1]$$ so $$\displaystyle p(2x−1)=p(y)=\sum_{n=0}^∞a_ny^n$$ converges.

## Exercise $$\PageIndex{9}$$

In the following exercises, suppose that $$\displaystyle p(x)=\sum_{n=0}^∞a_nx^n$$ satisfies $$\displaystyle \lim_{n→∞}\frac{a_{n+1}}{a_n}=1$$ where $$\displaystyle a_n≥0$$ for each $$\displaystyle n$$. State whether each series converges on the full interval $$\displaystyle (−1,1)$$, or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.

1. $$\displaystyle \sum_{n=0}^∞a_nx^{2n}$$

2. $$\displaystyle \sum_{n=0}^∞a_{2n}x^{2n}$$

Converges on $$\displaystyle (−1,1)$$ by the ratio test

3. $$\displaystyle \sum_{n=0}^∞a_{2n}x^n$$ (Hint:$$\displaystyle x=±\sqrt{x^2}$$)

4. $$\displaystyle \sum_{n=0}^∞a_{n^2}x^{n^2}$$ (Hint: Let $$\displaystyle b_k=a_k$$ if $$\displaystyle k=n^2$$ for some $$\displaystyle n$$, otherwise $$\displaystyle b_k=0$$.

Consider the series $$\displaystyle \sum b_kx^k$$ where $$\displaystyle b_k=a_k$$ if $$\displaystyle k=n^2$$ and $$\displaystyle b_k=0$$ otherwise. Then $$\displaystyle b_k≤a_k$$ and so the series converges on $$\displaystyle (−1,1)$$ by the comparison test.

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

## Exercise $$\PageIndex{10}$$

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

The approximation is more accurate near $$\displaystyle x=−1$$. The partial sums follow $$\displaystyle \frac{1}{1−x}$$ more closely as $$\displaystyle N$$ increases but are never accurate near $$\displaystyle x=1$$ since the series diverges there. 2. Plot the graphs of $$\displaystyle −ln(1−x)$$ and of the partial sums $$\displaystyle S_N=\sum_{n=1}^N\frac{x^n}{n}$$ for $$\displaystyle n=10,50,100$$ on the interval $$\displaystyle [−0.99,0.99]$$. Comment on the behavior of the sums near $$\displaystyle x=−1$$ and near $$\displaystyle x=1$$ as $$\displaystyle N$$ increases.

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

The approximation appears to stabilize quickly near both $$\displaystyle x=±1$$. 4. Plot the graphs of the partial sums $$\displaystyle S_N=\sum_{n=1}^Nsinnx^n$$ for $$\displaystyle n=10,50,100$$ on the interval $$\displaystyle [−0.99,0.99]$$. Comment on the behavior of the sums near $$\displaystyle x=−1$$ and near $$\displaystyle x=1$$ as $$\displaystyle N$$ increases.

5. Plot the graphs of the partial sums $$\displaystyle S_N=\sum_{n=0}^N(−1)^n\frac{x^{2n+1}}{(2n+1)!}$$ for $$\displaystyle n=3,5,10$$ on the interval $$\displaystyle [−2π,2π]$$. Comment on how these plots approximate $$\displaystyle sinx$$ as $$\displaystyle N$$ increases.

The polynomial curves have roots close to those of $$\displaystyle sinx$$ up to their degree and then the polynomials diverge from $$\displaystyle sinx$$. 6. Plot the graphs of the partial sums $$\displaystyle S_N=\sum_{n=0}^N(−1)^n\frac{x^{2n}}{(2n)!}$$ for $$\displaystyle n=3 ,5,10$$ on the interval $$\displaystyle [−2π,2π]$$. Comment on how these plots approximate $$\displaystyle cosx$$ as $$\displaystyle N$$ increases.