Chapter 9

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{4}}{2^{n}} $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots\) Hint: \(a_{n}=\frac{1}{n(n+1)}\)

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

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

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

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). \(\sin x, a=\frac{\pi}{6}\)

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

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

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{1}{2^{2}}+\frac{2}{3^{2}}+\frac{3}{4^{2}}+\frac{4}{5^{2}}+\cdots\)

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). $$ 0.36717171 \ldots $$

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

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

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

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

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

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{2}{1 \cdot 3 \cdot 4}+\frac{3}{2 \cdot 4 \cdot 5}+\frac{4}{3 \cdot 5 \cdot 6}+\frac{5}{4 \cdot 6 \cdot 7}+\cdots\)

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

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

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

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

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). \(\tan x, a=\frac{\pi}{4}\)

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{1}{1^{2}+1}+\frac{2}{2^{2}+1}+\frac{3}{3^{2}+1}+\frac{4}{4^{2}+1}+\cdots\)

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

$$ \text { Evaluate } \sum_{k=0}^{\infty}(-1)^{k} x^{k},-1

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. $$ \sum_{k=1}^{\infty} \frac{1}{1+4 k^{2}} $$

Find an explicit formula a \(a_{n}=\) ____ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ \frac{1}{2^{2}}, \frac{2}{2^{3}}, \frac{3}{2^{4}}, \frac{4}{2^{5}}, \ldots $$

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

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). \(1+x^{2}+x^{3}, a=1\)

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+\cdots\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{\cos n \pi}{n} $$

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

$$ \text { Show that } \sum_{k=1}^{\infty} \ln \frac{k}{k+1} \text { diverges. Hint: Obtain a formula } $$ $$ \text { for } S_{n} $$

Find an explicit formula a \(a_{n}=\) ____ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ -1, \frac{2}{3},-\frac{3}{5}, \frac{4}{7},-\frac{5}{9}, \ldots $$

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

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). \(2-x+3 x^{2}-x^{3}, a=-1\)

Determine convergence or divergence for each of the series. Indicate the test you use. \(3+\frac{3^{2}}{2 !}+\frac{3^{3}}{3 !}+\frac{3^{4}}{4 !}+\cdots\)

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

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

$$ \text { Show that } \sum_{k=2} \ln \left(1-\frac{1}{k^{2}}\right)=-\ln 2 $$

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

Let \(f(x)=\Sigma a_{n} x^{n}\) be an even function \((f(-x)=f(x))\) for \(x\) in \((-R, R) .\) Prove that \(a_{n}=0\) if \(n\) is odd. Hint: Use the Uniqueness Theorem.

Find the sum of each of the following series by recognizing how it is related to something familiar. (a) \(x-x^{2}+x^{3}-x^{4}+x^{5}-\cdots\) (b) \(\frac{1}{2 !}+\frac{x}{3 !}+\frac{x^{2}}{4 !}+\frac{x^{3}}{5 !}+\cdots\) (c) \(2 x+\frac{4 x^{2}}{2}+\frac{8 x^{3}}{3}+\frac{16 x^{4}}{4}+\cdots\)

Determine convergence or divergence for each of the series. Indicate the test you use. \(1+\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\cdots\)

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

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

A ball is dropped from a height of 100 feet. Each time it hits the floor, it rebounds to \(\frac{2}{3}\) its previous height. Find the total distance it travels before coming to rest.

Find an explicit formula a \(a_{n}=\) ____ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ 1, \frac{2}{2^{2}-1^{2}}, \frac{3}{3^{2}-2^{2}}, \frac{4}{4^{2}-3^{2}}, \ldots $$

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{\ln 2}{2^{2}}+\frac{\ln 3}{3^{2}}+\frac{\ln 4}{4^{2}}+\frac{\ln 5}{5^{2}}+\cdots\)

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