Chapter 9

Find the Maclaurin polynomial of order 4 for \(f(x)\) and use it to approximate \(f(0.12) .\) $$ f(x)=\tan ^{-1} x $$

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

find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series. $$ f(x)=\frac{x^{2}}{1-x^{4}} $$

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}\left(-\frac{3}{4}\right)^{n} $$

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

Use the Ratio Test to determine convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{n !}{n^{100}} $$

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

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=2}^{\infty} \frac{7}{4 k+2} $$

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

Find the Maclaurin polynomial of order 4 for \(f(x)\) and use it to approximate \(f(0.12) .\) $$ f(x)=\sinh x $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\frac{\cos x-1+x^{2} / 2}{x^{4}}\)

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

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n \sqrt{n}} $$

Find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{(x+1)^{n}}{n !} $$

Use the Ratio Test to determine convergence or divergence. $$ \sum_{n=1}^{\infty} n\left(\frac{1}{3}\right)^{n} $$

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 $$ \sum_{k=1}^{\infty} \frac{3}{k} $$

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=1}^{\infty} \frac{k^{2}}{e^{k}} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n \cos (n \pi)}{2 n-1} $$

Find the Taylor polynomial of order 3 based at a for the given function. $$ e^{x} ; a=1 $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\frac{1}{1-x} \cosh x\)

find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series. $$ f(x)=\int_{0}^{x} \ln (1+t) d t $$

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{2^{n}} $$

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

Use the Ratio Test to determine convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{n^{3}}{(2 n) !} $$

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 $$ \sum_{k=1}^{\infty} \frac{k !}{100^{k}} $$

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=1}^{\infty} \frac{3}{(4+3 k)^{7 / 6}} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{\cos (n \pi)}{n} $$

Find the Taylor polynomial of order 3 based at a for the given function. $$ \sin x ; a=\frac{\pi}{4} $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\frac{1}{1+x} \ln \left(\frac{1}{1+x}\right)=\frac{-\ln (1+x)}{1+x}\)

find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series. $$ f(x)=\int_{0}^{x} \tan ^{-1} t d t $$

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} $$

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

Use the Ratio Test to determine convergence or divergence. $$ \sum_{k=1}^{\infty} \frac{3^{k}+k}{k !} $$

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

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=1}^{\infty} \frac{1000 k^{2}}{1+k^{3}} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=e^{-n} \sin n $$

Find the Taylor polynomial of order 3 based at a for the given function. $$ \tan x ; a=\frac{\pi}{6} $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\frac{1}{1+x+x^{2}}\)

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

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n(n+1)} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n}{n+200} $$

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 $$ \sum_{k=1}^{\infty}\left(\frac{e}{\pi}\right)^{k+1} $$

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=1}^{\infty} k e^{-3 k^{2}} $$

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

Find the Taylor polynomial of order 3 based at a for the given function. $$ \sec x ; a=\frac{\pi}{4} $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\frac{1}{1-\sin x}\)

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{n}}{n !} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n !}{5+n} $$