Chapter 9

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

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.

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

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 \(\sin ^{-1} x\)

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

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.)

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

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

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 \(\sinh ^{-1} x\)

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

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

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

For the series given, determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002. $$ \sum_{k=1}^{\infty} \frac{1}{k^{3}} $$

Find an explicit formula a \(a_{n}=\) ____ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ -\frac{1}{3}, \frac{4}{9},-\frac{9}{27}, \frac{16}{81}, \ldots $$

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

Determine convergence or divergence for each of the series. Indicate the test you use. \(\sum_{n=1}^{\infty} \frac{4+\cos n}{n^{3}}\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{(-3)^{n+1}}{n^{2}} $$

For the series given, determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002. $$ \sum_{k=1}^{\infty} \frac{1}{1+k^{2}} $$

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

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

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

Determine convergence or divergence for each of the series. Indicate the test you use. \(\sum_{n=1}^{\infty} \frac{5^{2 n}}{n !}\)

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

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

For the series given, determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002. $$ \sum_{k=1}^{\infty} \frac{k}{e^{k^{2}}} $$

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

Find a good bound for the maximum value of the given expression, given that \(c\) is in the stated interval. Answers may vary depending on the technique used. (See Example 5.) $$ \left|\frac{4 c}{\sin c}\right| ;\left[\frac{\pi}{4}, \frac{\pi}{2}\right] $$

Find the power series representation of \(x /\left(x^{2}-3 x+2\right)\). Hint: Use partial fractions.

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

Find the radius of convergence of $$ \sum_{n=1}^{\infty} \frac{1 \cdot 2 \cdot 3 \cdots n}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} x^{2 n+1} $$

For the series given, determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002. $$ \sum_{k=1}^{\infty} \frac{k}{1+k^{4}} $$

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}\). Then use Theorem \(D\) to show that the sequence converges. $$ a_{n}=\frac{4 n-3}{2^{n}} $$

Find a good bound for the maximum value of the given expression, given that \(c\) is in the stated interval. Answers may vary depending on the technique used. (See Example 5.) $$ \left|\frac{4 c}{c+4}\right| ;[0,1] $$

Let \(f(x)=(1+x)^{1 / 2}+(1-x)^{1 / 2}\). Find the Maclaurin series for \(f\) and use it to find \(f^{(4)}(0)\) and \(f^{(51)}(0)\).

Let \(y=y(x)=x-\frac{\lambda}{3 !}+\frac{\lambda}{5 !}-\frac{\lambda}{7 !}+\cdots .\) Show that \(y\) satisfies the differential equation \(y^{\prime \prime}+y=0\) with the conditions \(y(0)=0\) and \(y^{\prime}(0)=1\). From this, guess a simple formula for \(y\).

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

Give an example of two series \(\sum a_{n}\) and \(\sum b_{n}\), both convergent, such that \(\sum a_{n} b_{n}\) diverges.

Find the radius of convergence of $$ \sum_{n=0}^{\infty} \frac{(p n) !}{(n !)^{p}} x^{n} $$ where \(p\) is a positive integer.

For the series given, determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002. $$ \sum_{k=1}^{\infty} \frac{1}{k(k+1)} $$

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}\). Then use Theorem \(D\) to show that the sequence converges. $$ a_{n}=\frac{n}{n+1}\left(2-\frac{1}{n^{2}}\right) $$

In each case, find the Maclaurin series for \(f(x)\) by use of known series and then use it to calculate \(f^{(4)}(0)\). (a) \(f(x)=e^{x+x^{2}}\) (b) \(f(x)=e^{\sin x}\) (c) \(f(x)=\int_{0}^{x} \frac{e^{t^{2}}-1}{t^{2}} d t\) (d) \(f(x)=e^{\cos x}=e \cdot e^{\cos x-1}\) (e) \(f(x)=\ln \left(\cos ^{2} x\right)\)

. Let \(\left\\{f_{n}\right\\}\) be the Fibonacci sequence defined by $$ f_{0}=0, \quad f_{1}=1, \quad f_{n+2}=f_{n+1}+f_{n} $$ (See Problem 52 of Section \(9.1\) and Problem 36 of Section 9.6.) If \(F(x)=\sum_{n=0}^{\infty} f_{n} x^{n}\), show that $$ F(x)-x F(x)-x^{2} F(x)=x $$ and then use this fact to obtain a simple formula for \(F(x)\).

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

Show that the positive terms of the alternating harmonic series form a divergent series. Show the same for the negative terms.

For what values of \(p\) does \(\sum_{n=2}^{\infty} 1 /\left[n(\ln n)^{p}\right]\) converge? Explain.

One can sometimes find a Maclaurin series by the method of equating coefficients. For example, let $$ \tan x=\frac{\sin x}{\cos x}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots $$ Then multiply by \(\cos x\) and replace \(\sin x\) and \(\cos x\) by their series to obtain $$ \begin{aligned} x-\frac{x^{3}}{6}+\cdots &=\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\left(1-\frac{x^{2}}{2}+\cdots\right) \\ &=a_{0}+a_{1} x+\left(a_{2}-\frac{a_{0}}{2}\right) x^{2}+\left(a_{3}-\frac{a_{1}}{2}\right) x^{3}+\cdots \end{aligned} $$ Thus, $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}-\frac{a_{0}}{2}=0, \quad a_{3}-\frac{a_{1}}{2}=-\frac{1}{6}, \quad \cdots $$ so $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}=0, \quad a_{3}=\frac{1}{3}, \quad \cdots $$ and therefore $$ \tan x=0+x+0+\frac{1}{3} x^{3}+\cdots $$ which agrees with Problem 1 . Use this method to find the terms through \(x^{4}\) in the series for \(\sec x\).

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

Suppose that \(\sum_{n=0}^{\infty} a_{n}(x-3)^{n}\) converges at \(x=-1 .\) Why can you conclude that it converges at \(x=6 ?\) Can you be sure that it converges at \(x=7 ?\) Explain.

Does \(\sum_{n=3}^{\infty} 1 /[n \cdot \ln n \cdot \ln (\ln n)]\) converge or diverge? Explain.