Chapter 9

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=2}^{\infty}\left(\frac{1}{k}-\frac{1}{k-1}\right) $$

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{(x+1)^{n}}{n !} $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{n \cos (n \pi)}{2 n-1}\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\frac{\cos x-1+x^{2} / 2}{x^{4}} $$

\(\sum_{k=1}^{\infty} \frac{k^{2}}{e^{k}}\)

\(\sum_{n=1}^{\infty} n\left(\frac{1}{3}\right)^{n}\)

In Problems 7–12, show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n \sqrt{n}} $$

In Problems 1-10, find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series (see Examples 1 and 2). $$ f(x)=\frac{x^{3}}{2-x^{3}} $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=1}^{\infty} \frac{3}{k} $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ \frac{x}{1 \cdot 2}-\frac{x^{2}}{2 \cdot 3}+\frac{x^{3}}{3 \cdot 4}-\frac{x^{4}}{4 \cdot 5}+\frac{x^{5}}{5 \cdot 6}-\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{\cos (n \pi)}{n}\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\frac{1}{1-x} \cosh x $$

\(\sum_{k=1}^{\infty} \frac{3}{(4+3 k)^{7 / 6}}\)

\(\sum_{n=1}^{\infty} \frac{n^{3}}{(2 n) !}\)

In Problems 7–12, show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{2^{n}} $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=1}^{\infty} \frac{k !}{100^{k}} $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ 1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=e^{-n} \sin n\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\frac{1}{1+x} \ln \left(\frac{1}{1+x}\right)=\frac{-\ln (1+x)}{1+x} $$

\(\sum_{k=1}^{\infty} \frac{3^{k}+k}{k !}\)

In Problems 7–12, show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=1}^{\infty} \frac{2}{(k+2) k} $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{e^{2 n}}{n^{2}+3 n-1}\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\frac{1}{1+x+x^{2}} $$

\(\sum_{k=1}^{\infty} k e^{-3 k^{2}}\)

In Problems \(11-34\), determine convergence or divergence for each of the series. Indicate the test you use. 11\. \(\sum_{n=1}^{\infty} \frac{n}{n+200}\)

In Problems 7–12, show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n(n+1)} $$

Obtain the power series in \(x\) for \(\ln [(1+x) /(1-x)]\) and specify its radius of convergence. Hint: $$ \ln [(1+x) /(1-x)]=\ln (1+x)-\ln (1-x) $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=1}^{\infty}\left(\frac{e}{\pi}\right)^{k+1} $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}-\frac{x^{10}}{10 !}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{e^{2 n}}{4^{n}}\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\frac{1}{1-\sin x} $$

\(\sum_{k=5}^{\infty} \frac{1000}{k(\ln k)^{2}}\)

\(\sum_{n=1}^{\infty} \frac{n !}{5+n}\)

In Problems 7–12, show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{n}}{n !} $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=1}^{\infty} \frac{4^{k+1}}{7^{k-1}} $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ x+2 x^{2}+3 x^{3}+4 x^{4}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{(-\pi)^{n}}{5^{n}}\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\sin ^{3} x $$

In Problems 13-22, use any test developed so far, including any from Section 9.2, to decide about the convergence or divergence of the series. Give a reason for your conclusion. \(\sum_{k=1}^{\infty} \frac{k^{2}+1}{k^{2}+5}\)

\(\sum_{n=1}^{\infty} \frac{n+3}{n^{2} \sqrt{n}}\)

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n} $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=2}^{\infty}\left(\frac{3}{(k-1)^{2}}-\frac{3}{k^{2}}\right) $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ x+2^{2} x^{2}+3^{2} x^{3}+4^{2} x^{4}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\left(\frac{1}{4}\right)^{n}+3^{n / 2}\)

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=x(\sin 2 x+\sin 3 x) $$

In Problems 13-22, use any test developed so far, including any from Section 9.2, to decide about the convergence or divergence of the series. Give a reason for your conclusion. \(\sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k}\)

\(\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}\)

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n^{1.1}} $$