
# 5.1E: Exercises


Exercise $$\PageIndex{1}$$

In exercises 1 - 4, 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 $$a_nx^n→0$$ as $$n→∞.$$

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

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

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

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

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

Exercise $$\PageIndex{2}$$

5) Suppose that $$\displaystyle \sum_{n=0}^∞a_n(x−3)^n$$ converges at $$x=6$$. At which of the following points must the series also converge? Use the fact that if $$\displaystyle \sum a_n(x−c)^n$$ 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. $$x=0.000001$$

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

6) Suppose that $$\displaystyle \sum_{n=0}^∞a_n(x+1)^n$$ converges at $$x=−2$$. At which of the following points must the series also converge? Use the fact that if $$\displaystyle \sum a_n(x−c)^n$$ 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. $$x=0.000001$$

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Exercise $$\PageIndex{3}$$

In the following exercises, suppose that $$\left|\dfrac{a_{n+1}}{a_n}\right|→1$$ as $$n→∞.$$ Find the radius of convergence for each series.

7) $$\displaystyle \sum_{n=0}^∞a_n2^nx^n$$

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

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

9) $$\displaystyle \sum_{n=0}^∞\frac{a_nπ^nx^n}{e^n}$$

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

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

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

$$\left|\dfrac{a_{n+1}(−1)^{n+1}x^{2n+2}}{a_n(−1)^nx^{2n}}\right| =|x^2|\left|\dfrac{a_{n+1}}{a_n}\right|→|x^2|$$ so $$R=1$$

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

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Exercise $$\PageIndex{4}$$

In exercises 13 - 22, find the radius of convergence $$R$$ and interval of convergence for $$\displaystyle \sum a_nx^n$$ with the given coefficients $$a_n$$.

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

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

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

15) $$\displaystyle \sum_{n=1}^∞\frac{nx^n}{2^n}$$

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

16) $$\displaystyle \sum_{n=1}^∞\frac{nx^n}{e^n}$$

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

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

18) $$\displaystyle \sum_{k=1}^∞\frac{k^ex^k}{e^k}$$

19) $$\displaystyle \sum_{k=1}^∞\frac{π^kx^k}{k^π}$$

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

20) $$\displaystyle \sum_{n=1}^∞\frac{x^n}{n!}$$

21) $$\displaystyle \sum_{n=1}^∞\frac{10^nx^n}{n!}$$

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

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

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Exercise $$\PageIndex{5}$$

In exercises 23 - 28, find the radius of convergence of each series.

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

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

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

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

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

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

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

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

28) $$\displaystyle \sum_{n=1}^∞\sin^2nx^n$$

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Exercise $$\PageIndex{6}$$

In exercises 29 - 32, use the ratio test to determine the radius of convergence of each series.

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

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

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

31) $$\displaystyle \sum_{n=1}^∞\frac{n!}{n^n}x^n$$

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

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

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Exercise $$\PageIndex{7}$$

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

33) $$f(x)=\dfrac{1}{x};a=1$$ (Hint: $$\dfrac{1}{x}=\dfrac{1}{1−(1−x)})$$

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

34) $$f(x)=\dfrac{1}{1−x^2};a=0$$

35) $$f(x)=\dfrac{x}{1−x^2};a=0$$

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

36) $$f(x)=\dfrac{1}{1+x^2};a=0$$

37) $$f(x)=\dfrac{x^2}{1+x^2};a=0$$

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

38) $$f(x)=\dfrac{1}{2−x};a=1$$

39) $$f(x)=\dfrac{1}{1−2x};a=0.$$

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

40) $$f(x)=\dfrac{1}{1−4x^2};a=0$$

41) $$f(x)=\dfrac{x^2}{1−4x^2};a=0$$

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

42) $$f(x)=\dfrac{x^2}{5−4x+x^2};a=2$$

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Exercise $$\PageIndex{8}$$

Use the result of exercise 43 to find the radius of convergence of the given series in the subsequent exercises (44 - 47).

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

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

44) $$\displaystyle \sum_{n=1}^∞\frac{x^n}{n^n}$$

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

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

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

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

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

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Exercise $$\PageIndex{9}$$

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

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

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

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

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

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

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Exercise $$\PageIndex{10}$$

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 $$a_n≥0$$ 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.

52) $$\displaystyle \sum_{n=0}^∞a_nx^{2n}$$

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

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

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

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

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

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Exercise $$\PageIndex{11}$$

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

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

The approximation is more accurate near $$x=−1$$. The partial sums follow $$\dfrac{1}{1−x}$$ more closely as $$N$$ increases but are never accurate near $$x=1$$ since the series diverges there.

58) [T] Plot the graphs of $$−\ln(1−x)$$ and of the partial sums $$\displaystyle S_N=\sum_{n=1}^N\frac{x^n}{n}$$ 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.

59) [T] Plot the graphs of the partial sums $$\displaystyle S_n=\sum_{n=1}^N\frac{x^n}{n^2}$$ 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.

The approximation appears to stabilize quickly near both $$x=±1$$.

60) [T] Plot the graphs of the partial sums $$\displaystyle S_N=\sum_{n=1}^N(\sin n) x^n$$ 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.

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

The polynomial curves have roots close to those of $$\sin x$$ up to their degree and then the polynomials diverge from $$\sin x$$.

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