5.1E: Exercises
- Page ID
- 91622
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 \(n→∞.\)
- Answer:
- 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 \((−R,R)\).
- Answer:
- 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. \(x=0.000001\)
- Answer:
- 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. \(x=0.000001\)
- Answer
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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) \(\displaystyle \sum_{n=0}^∞a_n2^nx^n\)
- Answer:
- \(\left|\dfrac{a_{n+1}2^{n+1}x^{n+1}}{a_n2^nx^n}\right| =2|x|\left|\dfrac{a_{n+1}}{a_n}\right|→2|x|\) so \(R=\frac{1}{2}\)
8) \(\displaystyle \sum_{n=0}^∞\frac{a_nx^n}{2^n}\)
9) \(\displaystyle \sum_{n=0}^∞\frac{a_nπ^nx^n}{e^n}\)
- Answer:
- \(\left|\dfrac{a_{n+1}(\dfrac{π}{e})^{n+1}x^{n+1}}{a_n(\dfrac{π}{e})^nx^n}\right| =\dfrac{π|x|}{e}\left|\dfrac{a_{n+1}}{a_n}\right|→\dfrac{π|x|}{e}\) so \(R=\frac{e}{π}\)
10) \(\displaystyle \sum_{n=0}^∞\frac{a_n(−1)^nx^n}{10^n}\)
11) \(\displaystyle \sum_{n=0}^∞a_n(−1)^nx^{2n}\)
- Answer:
- \(\left|\dfrac{a_{n+1}(−1)^{n+1}x^{2n+2}}{a_n(−1)^nx^{2n}}\right| =|x^2|\left|\dfrac{a_{n+1}}{a_n}\right|→|x^2|\) so \(R=1\)
12) \(\displaystyle \sum_{n=0}^∞a_n(−4)^nx^{2n}\)
- Answer
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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) \(\displaystyle \sum_{n=1}^∞\frac{(2x)^n}{n}\)
- Answer:
- \(a_n=\dfrac{2^n}{n}\) 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) \(\displaystyle \sum_{n=1}^∞\frac{nx^n}{2^n}\)
- Answer:
- \(a_n=\dfrac{n}{2^n}\) 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) \(\displaystyle \sum_{n=1}^∞\frac{n^2x^n}{2^n}\)
- Answer:
- \(a_n=\dfrac{n^2}{2^n}\) 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) \(\displaystyle \sum_{k=1}^∞\frac{π^kx^k}{k^π}\)
- Answer:
- \(a_k=\dfrac{π^k}{k^π}\) 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) \(\displaystyle \sum_{n=1}^∞\frac{10^nx^n}{n!}\)
- Answer:
- \(a_n=\dfrac{10^n}{n!},\dfrac{a_{n+1}x}{a_n}=\dfrac{10x}{n+1}→0<1\) so the series converges for all \(x\) by the ratio test and \(I=(−∞,∞)\).
22) \(\displaystyle \sum_{n=1}^∞(−1)^n\frac{x^n}{\ln(2n)}\)
- Answer
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Exercise \(\PageIndex{5}\)
In exercises 23 - 28, find the radius of convergence of each series.
23) \(\displaystyle \sum_{k=1}^∞\frac{(k!)^2x^k}{(2k)!}\)
- Answer:
- \(a_k=\dfrac{(k!)^2}{(2k)!}\) 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) \(\displaystyle \sum_{k=1}^∞\frac{k!}{1⋅3⋅5⋯(2k−1)}x^k\)
- Answer:
- \(a_k=\dfrac{k!}{1⋅3⋅5⋯(2k−1)}\) 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 \((^n_k)=\dfrac{n!}{k!(n−k)!}\)
- Answer:
- \(a_n=\dfrac{1}{(^{2n}_n)}\) 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) \(\displaystyle \sum_{n=1}^∞\sin^2nx^n\)
- Answer
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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) \(\displaystyle \sum_{n=1}^∞\frac{(n!)^3}{(3n)!}x^n\)
- Answer:
- \(\dfrac{a_{n+1}}{a_n}=\dfrac{(n+1)^3}{(3n+3)(3n+2)(3n+1)}→\dfrac{1}{27}\) so \(R=27\)
30) \(\displaystyle \sum_{n=1}^∞\frac{2^{3n}(n!)^3}{(3n)!}x^n\)
31) \(\displaystyle \sum_{n=1}^∞\frac{n!}{n^n}x^n\)
- Answer:
- \(a_n=\dfrac{n!}{n^n}\) 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) \(\displaystyle \sum_{n=1}^∞\frac{(2n)!}{n^{2n}}x^n\)
- Answer
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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: \(\dfrac{1}{x}=\dfrac{1}{1−(1−x)})\)
- Answer:
- \(\displaystyle f(x)=\sum_{n=0}^∞(1−x)^n\) on \(I=(0,2)\)
34) \(f(x)=\dfrac{1}{1−x^2};a=0\)
35) \(f(x)=\dfrac{x}{1−x^2};a=0\)
- Answer:
- \(\displaystyle \sum_{n=0}^∞x^{2n+1}\) on \(I=(−1,1)\)
36) \(f(x)=\dfrac{1}{1+x^2};a=0\)
37) \(f(x)=\dfrac{x^2}{1+x^2};a=0\)
- Answer:
- \(\displaystyle \sum_{n=0}^∞(−1)^nx^{2n+2}\) on \(I=(−1,1)\)
38) \(f(x)=\dfrac{1}{2−x};a=1\)
39) \(f(x)=\dfrac{1}{1−2x};a=0.\)
- Answer:
- \(\displaystyle \sum_{n=0}^∞2^nx^n\) on \(\left(−\frac{1}{2},\frac{1}{2}\right)\)
40) \(f(x)=\dfrac{1}{1−4x^2};a=0\)
41) \(f(x)=\dfrac{x^2}{1−4x^2};a=0\)
- Answer:
- \(\displaystyle \sum_{n=0}^∞4^nx^{2n+2}\) on \(\left(−\frac{1}{2},\frac{1}{2}\right)\)
42) \(f(x)=\dfrac{x^2}{5−4x+x^2};a=2\)
- Answer
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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 \(R=\frac{1}{r}\).
- Answer:
- \(|a_nx^n|^{1/n}=|a_n|^{1/n}|x|→|x|r\) 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) \(\displaystyle \sum_{k=1}^∞\left(\frac{k−1}{2k+3}\right)^kx^k\)
- Answer:
- \(a_k=\left(\dfrac{k−1}{2k+3}\right)^k\) 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) \(\displaystyle \sum_{n=1}^∞a_n=(n^{1/n}−1)^nx^n\)
- Answer:
- \(a_n=(n^{1/n}−1)^n\) so \((a_n)^{1/n}→0\) so \(R=∞\)
- Answer
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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 \(p(x)=−p(−x).\)
- Answer:
- 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 \(p(2x−1)\).
- Answer:
- If \(x∈[0,1],\) then \(y=2x−1∈[−1,1]\) so \(\displaystyle p(2x−1)=p(y)=\sum_{n=0}^∞a_ny^n\) converges.
- Answer
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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) \(\displaystyle \sum_{n=0}^∞a_{2n}x^{2n}\)
- Answer:
- 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 \(b_k=0\).)
- Answer:
- 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.
- Answer
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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.
- Answer:
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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.
- Answer:
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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.
- Answer:
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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.
- Answer
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Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.