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

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

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

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 \(f(x)\).

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

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10 n+1} $$

In Problems 15-20, 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). $$ 0.22222 \ldots $$

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. $$ 1+x+\frac{x^{2}}{\sqrt{2}}+\frac{x^{3}}{\sqrt{3}}+\frac{x^{4}}{\sqrt{4}}+\frac{x^{5}}{\sqrt{5}}+\cdots $$

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

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

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

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

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

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10 n^{1.1}+1} $$

In Problems 15-20, 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). $$ 0.21212121 \ldots $$

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. $$ 1-\frac{x}{1 \cdot 3}+\frac{x^{2}}{2 \cdot 4}-\frac{x^{3}}{3 \cdot 5}+\frac{x^{4}}{4 \cdot 6}-\cdots $$

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

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

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) \(f(2)\)

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

\(\sum_{n=1}^{\infty} \frac{4 n^{3}+3 n}{n^{5}-4 n^{2}+1}\)

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{1}{n \ln n} $$

In Problems 17-24, use the methods of Example 5 to find power series in \(x\) for each function \(f\). $$ f(x)=e^{-x} \cdot \frac{1}{1-x} $$

In Problems 15-20, 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). $$ 0.013013013 \ldots $$

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. $$ \frac{x}{2^{2}-1}+\frac{x^{2}}{3^{2}-1}+\frac{x^{3}}{4^{2}-1}+\frac{x^{4}}{5^{2}-1}+\cdots $$

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

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

Find the Maclaurin polynomial of order \(n\) ( \(n\) odd) for \(\sin x\). Then use it with \(n=5\) to approximate each of the following. (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) \(\sin (10)\)

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

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

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

In Problems 15-20, 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). $$ 0.125125125 \ldots $$

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. $$ 1-\frac{x}{2}+\frac{x^{2}}{2^{2}}-\frac{x^{3}}{2^{3}}+\frac{x^{4}}{2^{4}}-\cdots $$

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

In Problems 19-24, find the Taylor series in \(x-\) a through the term \((x-a)^{3}\). $$ e^{x}, a=1 $$

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

\(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots\) Hint: \(a_{n}=\frac{1}{n(n+1)}\)

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{4}}{2^{n}} $$

In Problems 15-20, 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). $$ 0.49999 \ldots $$

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. $$ 1+2 x+2^{2} x^{2}+2^{3} x^{3}+2^{4} x^{4}+\cdots $$

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 \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=(2 n)^{1 / 2 n}\)

In Problems 19-24, find the Taylor series in \(x-\) a through the term \((x-a)^{3}\). $$ \sin x, a=\frac{\pi}{6} $$

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

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{1}{\sqrt{n^{2}-1}} $$

In Problems 15-20, 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). $$ 0.36717171 \ldots $$

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. $$ 1+2 x+\frac{2^{2} x^{2}}{2 !}+\frac{2^{3} x^{3}}{3 !}+\frac{2^{4} x^{4}}{4 !}+\cdots $$

In Problems 21-30, find an explicit formula \(a_{n}=\) for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\)

In Problems 19-24, find the Taylor series in \(x-\) a through the term \((x-a)^{3}\). $$ \cos x, a=\frac{\pi}{3} $$

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