Chapter 10

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{n}{n+10} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=2+(0.1)^{n}$$

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

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=1 /(1-x)^{3}, \quad a=0\)

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

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

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

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? $$ \sum_{n=0}^{\infty}(-2)^{n}(n+1)(x-1)^{n} $$

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.) $$ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)} $$

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{n+(-1)^{n}}{n} $$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=e^{x}, \quad a=2\)

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

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

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

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

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

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=0}^{\infty} \frac{1}{n+4} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-2 n}{1+2 n} $$

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x} \end{equation}

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

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=2^{x}, \quad a=1\)

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

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

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

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

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{n}{n^{2}+3} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{2 n+1}{1-3 \sqrt{n}} $$

Use series to evaluate the limits. \begin{equation} \lim _{t \rightarrow 0} \frac{1-\cos t-\left(t^{2} / 2\right)}{t^{4}} \end{equation}

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}{1+\ln n}\end{equation}

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=\cos (2 x+(\pi / 2)), \quad a=\pi / 4\)

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

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

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? $$ \sum_{n=1}^{\infty} \frac{(4 x-5)^{2 n+1}}{n^{3 / 2}} $$

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\infty} \cos \frac{1}{n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-5 n^{4}}{n^{4}+8 n^{3}} $$

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.) $$ \sum_{n=3}^{\infty} \frac{(1 / n)}{(\ln n) \sqrt{\ln ^{2} n-1}} $$

Use series to evaluate the limits. \begin{equation} \lim _{\theta \rightarrow 0} \frac{\sin \theta-\theta+\left(\theta^{3} / 6\right)}{\theta^{5}} \end{equation}

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises \(29-34 .\) $$\cos ^{2} x \cdot \sin x$$

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

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=\sqrt{x+1}, \quad a=0\)

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-5)^{-n} $$

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

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? $$ \sum_{n=1}^{\infty} \frac{(3 x+1)^{n+1}}{2 n+2} $$

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=0}^{\infty} \frac{e^{n}}{e^{n}+n} $$

Use series to evaluate the limits. \begin{equation} \lim _{y \rightarrow 0} \frac{y-\tan ^{-1} y}{y^{3}} \end{equation}

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises \(29-34 .\) $$e^{\sin x}$$

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

Find the first three nonzero terms of the Maclaurin series for each function and the values of \(x\) for which the series converges absolutely. \(f(x)=\cos x-(2 /(1-x))\)

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(-100)^{n}}{n !} $$