Chapter 10

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=2}^{\infty} \frac{3^{n+2}}{\ln n}$$

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? $$ \sum_{n=0}^{\infty}(2 x)^{n} $$

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

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

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ \frac{5}{1 \cdot 2}+\frac{5}{2 \cdot 3}+\frac{5}{3 \cdot 4}+\dots+\frac{5}{n(n+1)}+\cdots $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{2^{n}-1}{2^{n}} $$

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1+x^{3}\right)^{-1 / 2} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\sin x, \quad a=\pi / 4\)

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2}(n+2) !}{n ! 3^{2 n}}$$

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{n}}{n^{2}} $$

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

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

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{n}{n^{2}+4} $$

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. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}} $$

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. $$ a_{1}=1, \quad a_{n+1}=a_{n}+\left(1 / 2^{n}\right) $$

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1+x^{2}\right)^{-1 / 3} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\tan x, \quad a=\pi / 4\)

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty} \frac{n 5^{n}}{(2 n+3) \ln (n+1)}$$

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{10^{1}}{(n+1) !} $$

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

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

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

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

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. $$ a_{1}=1, \quad a_{n+1}=a_{n} /(n+1) $$

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1+\frac{1}{x}\right)^{1 / 2} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\sqrt{x}, \quad a=4\)

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{7}{(2 n+5)^{n}}$$

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n}{10}\right)^{n} $$

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? $$ \sum_{n=1}^{\infty} \frac{x^{n}}{n \sqrt{n} 3^{n}} $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\begin{array}{l}{\sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3}} \\ {\text { (Hint: Limit Comparison with } \sum_{n=1}^{\infty}\left(1 / n^{2}\right) )}\end{array}\end{equation}

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{n^{2}}{e^{n / 3}} $$

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. $$ \sum_{n=1}^{\infty}\left(1-\frac{7}{4^{n}}\right) $$

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

Find the first four terms of the binomial series for the functions. \begin{equation} \frac{x}{\sqrt[3]{1+x}} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\sqrt{1-x}, \quad a=0\)

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

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=2}^{\infty}(-1)^{n+1} \frac{1}{\ln n} $$

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

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

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=2}^{\infty} \frac{n-4}{n^{2}-2 n+1} $$

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. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{5}{4^{n}} $$

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

Find the binomial series for the functions in Exercises \(11-14\) \begin{equation} (1+x)^{4} \end{equation}

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

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

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}\left(\frac{4 n+3}{3 n-5}\right)^{n}$$

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

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

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=2}^{\infty} \frac{n(n+1)}{\left(n^{2}+1\right)(n-1)}\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. $$ \sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right) $$