1.1E: Exercises

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 \(\displaystyle n→∞.\)

Answer

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 \(\displaystyle (−R,R)\).

Answer

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. \(\displaystyle x=0.000001\)

Answer

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. \(\displaystyle \sum_{n=0}^∞a_n2^nx^n\)

Answer

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

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

Answer

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

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

Answer

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

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

Answer

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

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

Answer

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

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

Answer

\(\displaystyle ∣\frac{a_{n+1}(-4)^{n+1}x^{2(n+1)}}{a_n(-4)^nx^{2n}}∣ =4|x^2|∣ \frac{a_{n+1}}{a_n}∣→4|x^2|\) 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. \(\displaystyle \sum_{n=1}^∞\frac{(2x)^n}{n}\)

Answer

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

Answer

R=1

Interval of convergence (-1,1)

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

Answer

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

Answer

\(\displaystyle a_n=\frac{n^2}{2^n}\) 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. \(\displaystyle \sum_{k=1}^∞\frac{π^kx^k}{k^π}\)

Answer:

\(\displaystyle a_k=\frac{π^k}{k^π}\) 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. \(\displaystyle \sum_{n=1}^∞\frac{10^nx^n}{n!}\)

Answer:

\(\displaystyle a_n=\frac{10^n}{n!},\frac{a_{n+1}x}{a_n}=\frac{10x}{n+1}→0<1\) 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. \(\displaystyle \sum_{k=1}^∞\frac{(k!)^2x^k}{(2k)!}\)

Answer

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

Answer

\(\displaystyle a_k=\frac{k!}{1⋅3⋅5⋯(2k−1)}\) 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 \(\displaystyle (^n_k)=\frac{n!}{k!(n−k)!}\)

Answer

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

Answer

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

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

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

Answer

\(\displaystyle a_n=\frac{n!}{n^n}\) 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: \(\displaystyle \frac{1}{x}=\frac{1}{1−(1−x)})\)

Answer

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

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

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

Answer

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

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

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

Answer

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

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

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

Answer

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

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

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

Answer

\(\displaystyle \sum_{n=0}^∞4^nx^{2n+2}\) 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 \(\displaystyle R=\frac{1}{r}\).

Answer

\(\displaystyle |a_nx^n|^{1/n}=|a_n|^{1/n}|x|→|x|r\) 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. \(\displaystyle \sum_{k=1}^∞(\frac{k−1}{2k+3})^kx^k\)

Answer

\(\displaystyle a_k=(\frac{k−1}{2k+3})^k\) 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. \(\displaystyle \sum_{n=1}^∞a_n=(n^{1/n}−1)^nx^n\)

Answer

\(\displaystyle a_n=(n^{1/n}−1)^n\) 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 \(\displaystyle p(x)=−p(−x).\)

Answer

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 \(\displaystyle p(2x−1)\).

Answer

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

Answer

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 \(\displaystyle b_k=0\).

Answer

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.

Answer

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.

Answer

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.

Answer

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.