Chapter 9

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 $$ \sum_{k=1}^{\infty} \frac{4^{k+1}}{7^{k-1}} $$

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=5}^{\infty} \frac{1000}{k(\ln k)^{2}} $$

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. $$ \cot ^{-1} x ; a=1 $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\sin ^{3} x\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n+3}{n^{2} \sqrt{n}} $$

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

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{(-\pi)^{n}}{5^{n}} $$

Find the Taylor polynomial of order 3 based at a for the given function. $$ \sqrt{x} ; a=2 $$

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=x(\sin 2 x+\sin 3 x)\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n^{1.1}} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1} $$

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 $$ \sum_{k=6}^{\infty} \frac{2}{k-5} $$

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. $$ \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} $$

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\left(\frac{1}{4}\right)^{n}+3^{n / 2} $$

Find the Taylor polynomial of order 3 based at 1 for \(f(x)=x^{3}-2 x^{2}+3 x+5\) and show that it is an exact representation of \(f(x)\).

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=x \sec \left(x^{2}\right)+\sin x\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10 n+1} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n^{2}}{n !} $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=2+(0.99)^{n} $$

Find the Taylor polynomial of order 4 based at 2 for \(f(x)=x^{4}\) and show that it represents \(f(x)\) exactly.

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10 n^{1.1}+1} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{\ln n}{2^{n}} $$

Write the given decimal as an infinite series, then find the sum of the series, and finally, use the result to write the decimal as a ratio of two integers (see Example 2). $$ 0.21212121 \ldots $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n^{100}}{e^{n}} $$

Find the Maclaurin polynomial of order \(n\) for \(f(x)=1 /(1-x)\). Then use it with \(n=4\) to approximate each of the following. (a) \(f(0.1)\) (b) \(f(0.5)\) (c) \(f(0.9)\) (d) \(f(2)\)

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=(1+x)^{3 / 2}\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{1}{n \ln n} $$

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{4 n^{3}+3 n}{n^{5}-4 n^{2}+1} $$

Write the given decimal as an infinite series, then find the sum of the series, and finally, use the result to write the decimal as a ratio of two integers (see Example 2). $$ 0.013013013 \ldots $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{\ln n}{\sqrt{n}} $$

Find the Maclaurin polynomial of order \(n(n\) odd \()\) for \(\sin x .\) Then use it with \(n=5\) to approximate each of the fol. lowing. (This example should convince you that the Maclaurin approximation can be exceedingly poor if \(x\) is far from zero. Compare your answers with those given by your calculator. What conclusion do you draw? (a) \(\sin (0.1)\) (b) \(\sin (0.5)\) (c) \(\sin (1)\) (d) \(\sin (10)\)

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, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=\left(1-x^{2}\right)^{2 / 3}\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n(1+\sqrt{n})} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n^{2}+1}{3^{n}} $$

Write the given decimal as an infinite series, then find the sum of the series, and finally, use the result to write the decimal as a ratio of two integers (see Example 2). $$ 0.125125125 \ldots $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{\ln (1 / n)}{\sqrt{2 n}} $$

Plot on the same axes the given function along with the Maclaurin polynomials of orders \(1,2,3\), and \(4 .\) $$ \cos 2 x $$

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). \(e^{x}, a=1\)