Chapter 10

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=1, \quad a_{n+1}=\frac{1+\tan ^{-1} n}{n} a_{n}$$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty}\left(\frac{1}{2^{1 / n}}-\frac{1}{2^{1 /(n+1)}}\right) $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{\sin ^{2} n}{2^{n}} $$

$$\sum_{n=1}^{\infty}(1 / \sqrt{n+1}) \quad diverges $$

The (second) second derivative test Use the equation $$f(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}\left(c_{2}\right)}{2}(x-a)^{2}$$ to establish the following test. \begin{equation} \begin{array}{l}{\text { Let } f \text { have continuous first and second derivatives and }} \\ {\text { suppose that } f^{\prime}(a)=0 . \text { Then }} \\ {\text { a. } f \text { has a local maximum at } a \text { if } f^{\prime \prime} \leq 0 \text { throughout an interval }} \\ {\text { whose interior contains } a ;} \\ {\text { b. } f \text { has a local minimum at } a \text { if } f^{\prime \prime} \geq 0 \text { throughout an interval }} \\\ {\text { whose interior contains a. }}\end{array} \end{equation}

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\tan ^{-1} n}{n^{1.1}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \frac{1}{4}-\frac{1}{6}+\frac{1}{8}-\frac{1}{10}+\frac{1}{12}-\frac{1}{14}+\cdots $$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty}\left(\frac{1}{\ln (n+2)}-\frac{1}{\ln (n+1)}\right) $$

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=\frac{1}{3}, \quad a_{n+1}=\frac{3 n-1}{2 n+5} a_{n}$$}$$

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) 1\(/(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\sec ^{-1} n}{n^{1.3}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ 1+\frac{1}{4}-\frac{1}{9}-\frac{1}{16}+\frac{1}{25}+\frac{1}{36}-\frac{1}{49}-\frac{1}{64}+\cdots $$

Find the sum of each series in Exercises \(41-48\) $$ \sum_{n=1}^{\infty}\left(\tan ^{-1}(n)-\tan ^{-1}(n+1)\right) $$

In Exercises \(41-48\) , use Theorem 20 to find the series' interval of convergence and, within this interval, the sum of the series as a function of \(x .\) $$ \sum_{n=0}^{\infty}\left(\frac{x^{2}-1}{2}\right)^{n} $$

\begin{equation} \begin{array}{l}{\text { a. Use Taylor's formula with } n=2 \text { to find the quadratic }} \\ {\text { approximation of } f(x)=(1+x)^{k} \text { at } x=0(k \text { a constant) }} \\ {\text { b. If } k=3, \text { for approximately what values of } x \text { in the interval }} \\ {[0,1] \text { will the error in the quadratic approximation be less }} \\ {\text { than } 1 / 100 ?}\end{array} \end{equation}

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\operatorname{coth} n}{n^{2}}\end{equation}

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n} $$

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=2, \quad a_{n+1}=\frac{2}{n} a_{n}$$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{2}}\right)^{n} $$

Estimate the value of \(\sum_{n=2}^{\infty}\left(1 /\left(n^{2}+4\right)\right)\) to within 0.1 of its exact value.

Improving approximations of \(\pi\) \begin{equation} \begin{array}{l}{\text { a. Let } P \text { be an approximation of } \pi \text { accurate to } n \text { decimals. Show }} \\ {\text { that } P+\sin P \text { gives an approximation correct to } 3 n \text { decimals. }} \\ {\text { (Hint: Let } P=\pi+x . )} \\ {\text { b. Try it with a calculator. }}\end{array} \end{equation}

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\tanh n}{n^{2}}\end{equation}

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{10^{n}} $$

Use a geometric series to represent each of the given functions as a power series about \(x=0,\) and find their intervals of convergence. $$ \text { a. } \quad f(x)=\frac{5}{3-x} \quad \text { b. } \quad g(x)=\frac{3}{x-2} $$

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=5, \quad a_{n+1}=\frac{\sqrt[n]{n}}{2} a_{n}$$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty}(\sqrt{2})^{n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{\ln n}{\ln 2 n} $$

How many terms of the convergent series \(\sum_{n=1}^{\infty}\left(1 / n^{1.1}\right)\) should be used to estimate its value with error at most 0.00001\(?\)

The Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is \(\sum_{n=0}^{\infty} a_{n} x^{n}\) A function defined by a power series \(\Sigma_{n=0}^{\infty} a_{n} x^{n}\) with a radius of convergence \(R>0\) has a Taylor series that converges to the function at every point of \((-R, R) .\) Show this by showing that the Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) itself. \begin{equation} \begin{array}{c}{\text { An immediate consequence of this is that series like }} \\ {x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}+\cdots} \\ {\text {and}}\end{array} \end{equation} \begin{equation} x^{2} e^{x}=x^{2}+x^{3}+\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\cdots \end{equation} obtained by multiplying Taylor series by powers of \(x,\) as well as series obtained by integration and differentiation of convergent power series, are themselves the Taylor series generated by the functions they represent.

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n \sqrt[n]{n}}\end{equation}

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=1, \quad a_{n+1}=\frac{1+\ln n}{n} a_{n}$$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{3}{2^{n}} $$

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{(0.01)^{n}}{n} $$

How many terms of the convergent series \(\sum_{n=4}^{\infty}\left(1 / n(\ln n)^{3}\right)\) should be used to estimate its value with error at most 0.01\(?\)

Taylor series for even functions and odd functions (Continuation of Section \(10.7,\) Exercise \(59 .\) ) Suppose that \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) converges for all \(x\) in an open interval \((-R, R) .\) Show that \begin{equation} \begin{array}{l}{\text { a. If } f \text { is even, then } a_{1}=a_{3}=a_{5}=\dots=0, \text { i.e., the Taylor }} \\ {\text { series for } f \text { at } x=0 \text { contains only even powers of } x .} \\ {\text { b. If } f \text { is odd, then } a_{0}=a_{4}=a_{4}=\cdots=0, \text { i.e., the Taylor }} \\ {\text { series for } f \text { at } x=0 \text { contains only odd powers of } x .}\end{array} \end{equation}

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\sqrt[n]{n}}{n^{2}}\end{equation}

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$ \frac{1}{1+t}=\sum_{n=0}^{\infty}(-1)^{n} t^{n}, \quad 0 < t < 1 $$

a. Find the interval of convergence of the power series $$\sum_{n=0}^{\infty} \frac{8}{4^{n+2}} x^{n}$$ b. Represent the power series in part (a) as a power series about \(x=3\) and identify the interval of convergence of the new series. (Later in the chapter you will understand why the new interval of convergence does not necessarily include all of the numbers in the original interval of convergence.)

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=\frac{1}{2}, \quad a_{n+1}=\frac{n+\ln n}{n+10} a_{n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=(0.03)^{1 / n} $$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}(-1)^{n+1} n $$

Replace \(x\) by \(-x\) in the Taylor series for \(\ln (1+x)\) to obtain a series for \(\ln (1-x)\) . Then subtract this from the Taylor series for \(\ln (1+x)\) to show that for \(|x|<1\) \begin{equation} \ln \frac{1+x}{1-x}=2\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots\right) \end{equation}

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{1+2+3+\cdots+n}\end{equation}

Determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n^{2}+3} $$

For what values of \(x\) does the series $$1-\frac{1}{2}(x-3)+\frac{1}{4}(x-3)^{2}+\cdots+\left(-\frac{1}{2}\right)^{n}(x-3)^{n}+\cdots$$ converge? What is its sum? What series do you get if you differentiate the given series term by term? For what values of \(x\) does the new series converge? What is its sum?

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} \cos \left(\frac{n \pi}{2}\right) $$

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\\ {x=0 .}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=\frac{1}{\sqrt{1+x}}, \quad|x| \leq \frac{3}{4}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(1+\frac{7}{n}\right)^{n} $$

How many terms of the Taylor series for \(\ln (1+x)\) should you add to be sure of calculating ln \((1.1)\) with an error of magnitude less than \(10^{-8} ?\) Give reasons for your answer.

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\\ {x=0 .}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(1+x)^{3 / 2},-\frac{1}{2} \leq x \leq 2$$