Chapter 10

Each of Exercises \(7-12\) gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$ a_{1}=a_{2}=1, \quad a_{n+2}=a_{n+1}+a_{n} $$

Find the binomial series for the functions. \begin{equation} \left(1+x^{2}\right)^{3} \end{equation}

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x^{2} \sin x$$

Find the Maclaurin series for the functions \(x e^{x}\)

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}\left(-\ln \left(e^{2}+\frac{1}{n}\right)\right)^{n+1}$$

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right) $$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !} $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{2^{n}}{3+4^{n}}\end{equation}

In Exercises \(7-14,\) write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$ \sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}-\frac{1}{3^{n}}\right) $$

Each of Exercises \(7-12\) gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$ a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} e^{-n} $$

Find the binomial series for the functions. \begin{equation} (1-2 x)^{3} \end{equation}

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\frac{x^{2}}{2}-1+\cos x$$

Find the Maclaurin series for the functions \(\frac{1}{1+x}\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sqrt{n}+1}{n+1} $$

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{-8}{(3+(1 / n))^{2 n}}$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty} \frac{4^{n} x^{2 n}}{n} $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}}\end{equation}

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{n}{n+1} $$

In Exercises \(7-14,\) write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}+\frac{(-1)^{n}}{5^{n}}\right) $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,-1,1,-1,1, \ldots\)

Find the binomial series for the functions. \begin{equation} \left(1-\frac{x}{2}\right)^{4} \end{equation}

Find the Maclaurin series for the functions \(\frac{2+x}{1-x}\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{3 \sqrt{n+1}}{\sqrt{n}+1} $$

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \sin ^{n}\left(\frac{1}{\sqrt{n}}\right)$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty} \frac{(x-1)^{n}}{n^{3} 3^{n}} $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty}\left(\frac{2 n+3}{5 n+4}\right)^{n}\end{equation}

In Exercises \(7-14,\) write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$ \sum_{n=0}^{\infty}\left(\frac{2^{n+1}}{5^{n}}\right) $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(-1,1,-1,1,-1, \ldots\)

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\sin x-x+\frac{x^{3}}{3 !}$$

In Exercises \(15-18\) , use series to estimate the integrals' values with an error of magnitude less than \(10^{-5}\) . (The answer section gives the integrals' values rounded to seven decimal places.) \begin{equation} \int_{0}^{0.6} \sin x^{2} d x \end{equation}

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x \cos \pi x$$

Find the Maclaurin series for the functions \(\sin 3 x\)

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1}(0.1)^{n} $$

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{1}{n}\right)^{n^{2}}$$ $$(Hint: \lim _{n \rightarrow \infty}(1+x / n)^{n}=e^{x} )$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{x^{n}}{\sqrt{n^{2}+3}} $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\begin{array}{l}{\sum_{n=2}^{\infty} \frac{1}{\ln n}} \\\ {\text {(Hint: Limit Comparison with } \sum_{n=2}^{\infty}(1 / n) )}\end{array}\end{equation}

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} $$

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ 1+\left(\frac{2}{5}\right)+\left(\frac{2}{5}\right)^{2}+\left(\frac{2}{5}\right)^{3}+\left(\frac{2}{5}\right)^{4}+\cdots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,-4,9,-16,25, \dots\)

Use series to estimate the integrals' values with an error of magnitude less than \(10^{-5}\) . (The answer section gives the integrals' values rounded to seven decimal places.) \begin{equation} \int_{0}^{0.4} \frac{e^{-x}-1}{x} d x \end{equation}

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x^{2} \cos \left(x^{2}\right)$$

Find the Maclaurin series for the functions \(\sin \frac{x}{2}\)

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{(0.1)^{n}}{n} $$

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n^{1+n}}$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n+1}}{\sqrt{n}+3} $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\begin{array}{l}{\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n^{2}}\right)} \\ {\text { (Hint: Limit Comparison with } \sum_{n=1}^{\infty}\left(1 / n^{2}\right) )}\end{array}\end{equation}

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ 1+(-3)+(-3)^{2}+(-3)^{3}+(-3)^{4}+\cdots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}, \dots\)

Use series to estimate the integrals' values with an error of magnitude less than \(10^{-5}\) . (The answer section gives the integrals' values rounded to seven decimal places.) \begin{equation} \int_{0}^{0.5} \frac{1}{\sqrt{1+x^{4}}} d x \end{equation}