Chapter 10

Use the Integral Test to determine if the series in Exercises \(1-12\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=2}^{\infty} \frac{1}{5 n+10 \sqrt{n}} $$

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

In Exercises \(9-16,\) use the Limit Comparison Test to determine if each series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{3+4^{n}} $$

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? $$ \sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !} $$

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

In Exercises \(1 - 14 ,\) determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \ln \left( 1 + \frac { 1 } { n } \right) $$

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. $$ a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n} $$

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. $$\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}-\frac{1}{3^{n}}\right)$$

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

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$x e^{x}$$

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

In Exercises \(9-16,\) use the Limit Comparison Test to determine if each series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}} $$

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? $$ \sum_{n=1}^{\infty} \frac{4^{n} x^{2 n}}{n} $$

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 \(1 - 14 ,\) determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { \sqrt { n } + 1 } { n + 1 } $$

Which of the series in Exercises 13 46 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.) $$ \sum_{n=1}^{\infty} \frac{1}{10^{n}} $$

Find a formula for the \(n\)th term of the sequence. $$ 1,-1,1,-1,1, \ldots $$

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. $$\sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}+\frac{(-1)^{n}}{5^{n}}\right)$$

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

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$x^{2} \sin x$$

Which of the series in Exercises 13 46 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}\left(1-\frac{x}{2}\right)^{4}\end{equation}

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? $$ \sum_{n=1}^{\infty} \frac{(x-1)^{n}}{n^{3} 3^{n}} $$

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 \(1 - 14 ,\) determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 3 \sqrt { n + 1 } } { \sqrt { n } + 1 } $$

Find a formula for the \(n\)th term of the sequence. $$ -1,1,-1,1,-1, \dots $$

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. $$\sum_{n=0}^{\infty}\left(\frac{2^{n+1}}{5^{n}}\right)$$

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

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$\frac{x^{2}}{2}-1+\cos x$$

Which of the series in Exercises 13 46 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 \(9-16,\) use the Limit Comparison Test to determine if each series converges or diverges. $$ \begin{array}{l}{\sum_{n=2}^{\infty} \frac{1}{\ln n}} \\ {\text {(Hint: Limit Comparison with } \sum_{n=2}^{\infty}(1 / n) )}\end{array} $$

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? $$ \sum_{n=0}^{\infty} \frac{x^{n}}{\sqrt{n^{2}+3}} $$

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

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } ( 0.1 ) ^ { n } $$

Find a formula for the \(n\)th term of the sequence. $$ 1,-4,9,-16,25, \dots $$

In Exercises \(15-22\) , determine if the geometric series converges or diverges. If a series converges, find its sum. $$1+\left(\frac{2}{5}\right)+\left(\frac{2}{5}\right)^{2}+\left(\frac{2}{5}\right)^{3}+\left(\frac{2}{5}\right)^{4}+\cdots$$

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

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$\sin x-x+\frac{x^{3}}{3 !}$$

In Exercises \(9-16,\) use the Limit Comparison Test to determine if each series converges or diverges. $$ \begin{array}{l}{\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n^{2}}\right)} \\\ {\text {(Hint: Limit Comparison with } \sum_{n=1}^{\infty}\left(1 / n^{2}\right) )}\end{array} $$

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? $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n+1}}{\sqrt{n}+3} $$

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

Which of the series in Exercises 13 46 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.) $$ \sum_{n=1}^{\infty} \frac{5}{n+1} $$

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { ( 0.1 ) ^ { n } } { n } $$

Find a formula for the \(n\)th term of the sequence. $$ 1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}, \ldots $$

In Exercises \(15-22\) , determine if the geometric series converges or diverges. If a series converges, find its sum. $$1+(-3)+(-3)^{2}+(-3)^{3}+(-3)^{4}+\cdots$$

Find the Maclaurin series for the functions. 7 \(\cos (-x)\)

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$x \cos \pi x$$

Which of the series in Exercises 13 46 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.) $$ \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} $$

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}

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? $$ \sum_{n=0}^{\infty} \frac{n(x+3)^{n}}{5^{n}} $$