Chapter 9

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

In Problems 17-24, use the methods of Example 5 to find power series in \(x\) for each function \(f\). $$ f(x)=\left(\tan ^{-1} x\right)\left(1+x^{2}+x^{4}\right) $$

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(\frac{1}{2^{2}}, \frac{2}{2^{3}}, \frac{3}{2^{4}}, \frac{4}{2^{5}}, \ldots\)

In Problems 19-24, find the Taylor series in \(x-\) a through the term \((x-a)^{3}\). $$ \tan x, a=\frac{\pi}{4} $$

$$ \cos (x-\pi) $$

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

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

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(-1, \frac{2}{3},-\frac{3}{5}, \frac{4}{7},-\frac{5}{9}, \ldots\)

In Problems 19-24, find the Taylor series in \(x-\) a through the term \((x-a)^{3}\). $$ 1+x^{2}+x^{3}, a=1 $$

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{\cos n \pi}{n} $$

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

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(1, \frac{1}{1-\frac{1}{2}}, \frac{1}{1-\frac{2}{3}}, \frac{1}{1-\frac{3}{4}}, \ldots\)

In Problems 19-24, find the Taylor series in \(x-\) a through the term \((x-a)^{3}\). $$ 2-x+3 x^{2}-x^{3}, a=-1 $$

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

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

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. $$ 1+\frac{x+1}{2}+\frac{(x+1)^{2}}{2^{2}}+\frac{(x+1)^{3}}{2^{3}}+\cdots $$

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(1, \frac{2}{2^{2}-1^{2}}, \frac{3}{3^{2}-2^{2}}, \frac{4}{4^{2}-3^{2}}, \ldots\)

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

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) \(2 x+\frac{4 x^{2}}{2}+\frac{8 x^{3}}{3}+\frac{16 x^{4}}{4}+\cdots\)

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.

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(\frac{1}{2-\frac{1}{2}}, \frac{2}{3-\frac{1}{3}}, \frac{3}{4-\frac{1}{4}}, \frac{4}{5-\frac{1}{5}}, \ldots\)

\(\frac{\ln 2}{2^{2}}+\frac{\ln 3}{3^{2}}+\frac{\ln 4}{4^{2}}+\frac{\ln 5}{5^{2}}+\cdots\)

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

Three people, A, B, and C, divide an apple as follows. First they divide it into fourths, each taking a quarter. Then they divide the leftover quarter into fourths, each taking a quarter, and so on. Show that each gets a third of the apple.

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(\sin 1,2 \sin \frac{1}{2}, 3 \sin \frac{1}{3}, 4 \sin \frac{1}{4}, \ldots\)

Recall that $$ \sin ^{-1} x=\int_{0}^{x} \frac{1}{\sqrt{1-t^{2}}} d t $$ Find the first four nonzero terms in the Maclaurin series for \(\sin ^{-1} x\).

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

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

Suppose that the government pumps an extra \(\$ 1\) billion into the economy. Assume that each business and individual saves \(25 \%\) of its income and spends the rest, so of the initial \(\$ 1\) billion, \(75 \%\) is respent by individuals and businesses. Of that amount, \(75 \%\) is spent, and so forth. What is the total increase in spending due to the government action? (This is called the multiplier effect in economics.)

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. $$ (x+3)-2(x+3)^{2}+3(x+3)^{3}-4(x+3)^{4}+\cdots $$

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(-\frac{1}{3}, \frac{4}{9},-\frac{9}{27}, \frac{16}{81}, \ldots\)

Given that $$ \sinh ^{-1} x=\int_{0}^{x} \frac{1}{\sqrt{1+t^{2}}} d t $$ find the first four nonzero terms in the Maclaurin series for \(\sinh ^{-1} x\)

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

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

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(2,1, \frac{2^{3}}{3^{2}}, \frac{2^{4}}{4^{2}}, \frac{2^{5}}{5^{2}}, \ldots\)

Calculate, accurate to four decimal places, $$ \int_{0}^{1} \cos \left(x^{2}\right) d x $$

$$ \left|e^{2 c}+e^{-2 c}\right| ;[0,3] $$

\(\sum_{n=1}^{\infty} \frac{4+\cos n}{n^{3}}\)

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{(-3)^{n+1}}{n^{2}} $$

Let \(k\) be an arbitrary number and \(-1

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 \(\lim _{n \rightarrow \infty} a_{n}\). \(1-\frac{1}{2}, \frac{1}{2}-\frac{1}{3}, \frac{1}{3}-\frac{1}{4}, \frac{1}{4}-\frac{1}{5}, \ldots\)

Calculate, accurate to five decimal places, $$ \int_{0}^{0.5} \sin \sqrt{x} d x $$

\(\sum_{n=1}^{\infty} \frac{5^{2 n}}{n !}\)

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

Suppose that \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n}\) for \(|x|