Chapter 10

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

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? $$ \sum_{n=1}^{\infty} \frac{1}{2 \cdot 4 \cdot 6 \cdots(2 n)} x^{n} $$

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

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

Use series to evaluate the limits. \begin{equation} \lim _{y \rightarrow 0} \frac{\tan ^{-1} y-\sin y}{y^{3} \cos y} \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{\sqrt{n}}{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. \(f(x)=\left(1-x+x^{2}\right) e^{x}\)

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

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} e^{-n}\left(n^{3}\right)$$

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? $$ \sum_{n=1}^{\infty} \frac{3 \cdot 5 \cdot 7 \cdots(2 n+1)}{n^{2} \cdot 2^{n}} x^{n+1} $$

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

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. $$ \sum_{n=0}^{\infty} \cos n \pi $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-n^{3}}{70-4 n^{2}} $$

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow \infty} x^{2}\left(e^{-1 / x^{2}}-1\right) \end{equation}

Estimate the error if \(P_{3}(x)=x-\left(x^{3} / 6\right)\) is used to estimate the value of \(\sin x\) at \(x=0.1 .\)

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-n}{n 2^{n}} \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. \(f(x)=(\sin x) \ln (1+x)\)

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

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(n+3) !}{3 ! n ! 3^{n}}$$

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) $$

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

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

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

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow \infty}(x+1) \sin \frac{1}{x+1} \end{equation}

Estimate the error if \(P_{4}(x)=1+x+\left(x^{2} / 2\right)+\left(x^{3} / 6\right)+\left(x^{4} / 24\right)\) is used to estimate the value of \(e^{x}\) at \(x=1 / 2.\)

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

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

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}$$

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

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}\left(\frac{3}{n^{2}}-\frac{3}{(n+1)^{2}}\right) $$

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow 0} \frac{\ln \left(1+x^{2}\right)}{1-\cos 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{1}{3^{n-1}+1}\end{equation}

Use the Taylor series generated by \(e^{x}\) at \(x=a\) to show that $$e^{x}=e^{a}\left[1+(x-a)+\frac{(x-a)^{2}}{2 !}+\cdots\right].$$

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

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.) $$ \sum_{n=1}^{\infty} \frac{8 \tan ^{-1} n}{1+n^{2}} $$

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

In Exercises \(37-40,\) find the series' radius of convergence. $$ \sum_{n=1}^{\infty} \frac{n !}{3 \cdot 6 \cdot 9 \cdot \cdot \cdot 3 n} x^{n} $$

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}(\ln \sqrt{n+1}-\ln \sqrt{n}) $$

For approximately what values of \(x\) can you replace sin \(x\) by \(x-\left(x^{3} / 6\right)\) with an error of magnitude no greater than \(5 \times 10^{-4} ?\) Give reasons for your answer.

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow 2} \frac{x^{2}-4}{\ln (x-1)} \end{equation}

If \(\cos x\) is replaced by \(1-\left(x^{2} / 2\right)\) and \(|x|<0.5,\) what estimate can be made of the error? Does \(1-\left(x^{2} / 2\right)\) tend to be too large, or too small? Give reasons for your answer.

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-1}+1}{3^{n}}\end{equation}

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

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

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

In Exercises \(37-40,\) find the series' radius of convergence. $$ \sum_{n=1}^{\infty}\left(\frac{2 \cdot 4 \cdot 6 \cdots(2 n)}{2 \cdot 5 \cdot 8 \cdots(3 n-1)}\right)^{2} x^{n} $$

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}(\tan (n)-\tan (n-1)) $$

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow 0} \frac{\sin 3 x^{2}}{1-\cos 2 x} \end{equation}

How close is the approximation \(\sin x=x\) when \(|x|<10^{-3} ?\) For which of these values of \(x\) is \(x<\sin x ?\)