Chapter 10

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\cos ^{2} x\left(\operatorname{Hin} t \cdot \cos ^{2} x=(1+\cos 2 x) / 2\right)$$

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

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

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^{n}}$$

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

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{2 \sqrt{n}+\sqrt[3]{n}}\end{equation}

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ \left(\frac{1}{8}\right)+\left(\frac{1}{8}\right)^{2}+\left(\frac{1}{8}\right)^{3}+\left(\frac{1}{8}\right)^{4}+\left(\frac{1}{8}\right)^{5}+\cdots $$

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.) $$ \sum_{n=1}^{\infty}-\frac{1}{8^{n}} $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. $$ \frac{1}{9}, \frac{2}{12}, \frac{2^{2}}{15}, \frac{2^{3}}{18}, \frac{2^{4}}{21}, \dots $$

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}^{0.35} \sqrt[3]{1+x^{2}} d x \end{equation}

Find the Maclaurin series for the functions 5 \(\cos \pi x\)

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

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty}(-1)^{n} n^{2} e^{-n}$$

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? $$ \sum_{n=0}^{\infty} \frac{n x^{n}}{4^{n}\left(n^{2}+1\right)} $$

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{3}{n+\sqrt{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.) $$ \sum_{n=1}^{\infty} \frac{-8}{n} $$

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ \left(\frac{-2}{3}\right)^{2}+\left(\frac{-2}{3}\right)^{3}+\left(\frac{-2}{3}\right)^{4}+\left(\frac{-2}{3}\right)^{5}+\left(\frac{-2}{3}\right)^{6}+\cdots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. $$ -\frac{3}{2},-\frac{1}{6}, \frac{1}{12}, \frac{3}{20}, \frac{5}{30}, \dots $$

Use series to approximate the values of the integrals in Exercises \(19-22\) with an error of magnitude less than \(10^{-8}\) . \begin{equation} \int_{0}^{0.1} \frac{\sin x}{x} d x \end{equation}

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

Find the Maclaurin series for the functions \(\cosh x=\frac{e^{x}+e^{-x}}{2}\)

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{3}+1} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} n !(-e)^{-n}$$

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

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\sin ^{2} n}{2^{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.) $$ \sum_{n=2}^{\infty} \frac{\ln n}{n} $$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 0 . \overline{23}=0.232323 \ldots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(0,3,8,15,24, \dots\)

Use series to approximate the values of the integrals with an error of magnitude less than \(10^{-8}\) . \begin{equation} \int_{0}^{0.1} e^{-x^{2}} d x \end{equation}

In Exercises \(17-44\) , use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n !}{10^{n}}$$

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

Find the Maclaurin series for the functions \(\sinh x=\frac{e^{x}-e^{-x}}{2}\)

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n !}{2^{n}} $$

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? $$ \sum_{n=1}^{\infty} \sqrt[n]{n}(2 x+5)^{n} $$

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1+\cos n}{n^{2}}\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.) $$ \sum_{n=2}^{\infty} \frac{\ln n}{\sqrt{n}} $$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 0 . \overline{234}=0.234234234 \ldots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(-3,-2,-1,0,1, \ldots\)

Use series to approximate the values of the integrals with an error of magnitude less than \(10^{-8}\) . \begin{equation} \int_{0}^{0.1} \sqrt{1+x^{4}} d x \end{equation}

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

Find the Maclaurin series for the functions \(x^{4}-2 x^{3}-5 x+4\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{2 n}{3 n-1}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+3} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n !}{10^{n}}$$

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? $$ \sum_{n=1}^{\infty}\left(2+(-1)^{n}\right) \cdot(x+1)^{n-1} $$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 0 . \overline{7}=0.7777 \ldots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,5,9,13,17, \dots\)

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.) $$ \sum_{n=1}^{\infty} \frac{2^{n}}{3^{n}} $$

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\frac{2}{(1-x)^{3}}$$

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