Chapter 9

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

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{n}{3 n-1}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\tan x $$

\(\sum_{k=0}^{\infty} \frac{1}{k+3}\)

In Problems \(1-4\), use the Limit Comparison Test to determine convergence or divergence. 1\. \(\sum_{n=1}^{\infty} \frac{n}{n^{2}+2 n+3}\)

In Problems 1–6, show that each alternating series converges, and then estimate the error made by using the partial sum as an approximation to the sum S of the series (see Examples 1–3). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2}{3 n+1} $$

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). $$ f(x)=\frac{1}{1+x} $$

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

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{3 n+2}{n+1}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\tanh x $$

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

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. $$ \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{-k-2} $$

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{4 n^{2}+2}{n^{2}+3 n-1}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=e^{x} \sin x $$

\(\sum_{k=0}^{\infty} \frac{k}{k^{2}+3}\)

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

In Problems 1–6, show that each alternating series converges, and then estimate the error made by using the partial sum as an approximation to the sum S of the series (see Examples 1–3). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\ln (n+1)} $$

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). $$ f(x)=\frac{1}{(1-x)^{3}} $$

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. $$ \sum_{k=0}^{\infty}\left[2\left(\frac{1}{4}\right)^{k}+3\left(-\frac{1}{5}\right)^{k}\right] $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{3 n^{2}+2}{2 n-1}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=e^{-x} \cos x $$

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

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

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). $$ f(x)=\frac{x}{(1+x)^{2}} $$

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. $$ \sum_{k=1}^{\infty}\left[5\left(\frac{1}{2}\right)^{k}-3\left(\frac{1}{7}\right)^{k+1}\right] $$

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{n}}{n^{2}} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{n^{3}+3 n^{2}+3 n}{(n+1)^{3}}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=(\cos x) \ln (1+x) $$

$$ f(x)=\ln (1+x) $$

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

In Problems 5-10, use the Ratio Test to determine convergence or divergence. 5\. \(\sum_{n=1}^{\infty} \frac{8^{n}}{n !}\)

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). $$ f(x)=\frac{1}{2-3 x}=\frac{\frac{1}{2}}{1-\frac{3}{2} x} $$

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. $$ \sum_{k=1}^{\infty} \frac{k-5}{k+2} $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{\sqrt{3 n^{2}+2}}{2 n+1}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=(\sin x) \sqrt{1+x} $$

\(\sum_{k=100}^{\infty} \frac{3}{(k+2)^{2}}\)

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

In Problems 1–6, show that each alternating series converges, and then estimate the error made by using the partial sum as an approximation to the sum S of the series (see Examples 1–3). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\ln n}{\sqrt{n}} $$

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). $$ f(x)=\frac{1}{3+2 x} $$

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. $$ \sum_{k=1}^{\infty}\left(\frac{9}{8}\right)^{k} $$

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{(x-2)^{n}}{n} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=(-1)^{n} \frac{n}{n+2}\)

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, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=e^{x}+x+\sin x $$

\(\sum_{n=1}^{\infty} \frac{n !}{n^{100}}\)

In Problems 7–12, show that each series converges absolutely. $$ \sum_{n=1}^{\infty}\left(-\frac{3}{4}\right) n $$

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). $$ f(x)=\frac{x^{2}}{1-x^{4}} $$