Chapter 10

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^{2}+3^{2}+\cdots+n^{2}}\end{equation}

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

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. $$a_{1}=\frac{1}{2}, \quad a_{n+1}=\left(a_{n}\right)^{n+1}$$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} \frac{\cos n \pi}{5^{n}} $$

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

Logarithmic \(p\) -series a. Show that the improper integral $$\int_{2}^{\infty} \frac{d x}{x(\ln x)^{p}} \quad(p \text { a positive constant })$$ converges if and only if \(p>1\) b. What implications does the fact in part (a) have for the convergence of the series $$ \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}} ? $$ Give reasons for your answer.

According to the Alternating Series Estimation Theorem, how many terms of the Taylor series for tan \(^{-1} 1\) would you have to add to be sure of finding \(\pi / 4\) with an error of magnitude less than \(10^{-3} ?\) Give reasons for your answer.

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

The series $$\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\frac{x^{11}}{11 !}+\cdots$$ converges to sin \(x\) for all \(x\) . a. Find the first six terms of a series for cos \(x\) . For what values of \(x\) should the series converge? b. By replacing \(x\) by 2\(x\) in the series for \(\sin x,\) find a series that converges to \(\sin 2 x\) for all \(x .\) c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for 2 \(\sin x \cos x\) . Compare your answer with the answer in part (b).

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{2^{n} n ! n !}{(2 n) !}$$

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

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sqrt[n]{10 n} $$

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} $$f(x)=\frac{x}{x^{2}+1}, \quad|x| \leq 2$$

Show that the Taylor series for \(f(x)=\tan ^{-1} x\) diverges for \(|x|>1 .\)

If \(\sum_{n=1}^{\infty} a_{n}\) is a convergent series of nonnegative numbers, can anything be said about \(\sum_{n=1}^{\infty}\left(a_{n} / n\right) ?\) Explain.

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

The series $$e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}+\cdots$$ converges to \(e^{x}\) for all \(x .\) a. Find a series for \((d / d x) e^{x} .\) Do you get the series for \(e^{x} ?\) Explain your answer. b. Find a series for \(\int e^{x} d x .\) Do you get the series for \(e^{x} ?\) Explain your answer. c. Replace \(x\) by \(-x\) in the series for \(e^{x}\) to find a series that con- verges to \(e^{-x}\) for all \(x .\) Then multiply the series for \(e^{x}\) and \(e^{-x}\) to find the first six terms of a series for \(e^{-x} \cdot e^{x} .\)

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}(3 n) !}{n !(n+1) !(n+2) !}$$

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

Euler's constant Graphs like those in Figure 10.11 suggest that as \(n\) increases there is little change in the difference between the sum $$ 1+\frac{1}{2}+\dots+\frac{1}{n} $$ and the integral $$ \ln n=\int_{1}^{n} \frac{1}{x} d x $$ To explore this idea, carry out the following steps. a. By taking \(f(x)=1 / x\) in the proof of Theorem \(9,\) show that $$ \ln (n+1) \leq 1+\frac{1}{2}+\cdots+\frac{1}{n} \leq 1+\ln n $$ or $$ 0<\ln (n+1)-\ln n \leq 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \leq 1 $$ Thus, the sequence $$ a_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n $$ is bounded from below and from above. b. Show that $$ \frac{1}{n+1}<\int_{n}^{n+1} \frac{1}{x} d x=\ln (n+1)-\ln n $$ and use this result to show that the sequence \(\left\\{a_{n}\right\\}\) in part (a) is decreasing. since a decreasing sequence that is bounded from below converges, the numbers \(a_{n}\) defined in part (a) converge: $$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \rightarrow \gamma $$ The number \(\gamma,\) whose value is 0.5772\(\ldots\) is called Euler's constant.

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} $$f(x)=e^{-x} \cos 2 x, \quad|x| \leq 1$$

Suppose that \(a_{n}>0\) and \(b_{n}>0\) for \(n \geq N(N\) an integer). If \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=\infty\) and \(\sum a_{n}\) converges, can anything be said about \(\sum b_{n}^{?}\) Give reasons for your answer.

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(n !)^{n}}{\left(n^{n}\right)^{2}}$$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{2}{10^{n}} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(\frac{3}{n}\right)^{1 / n} $$

Estimating Pi About how many terms of the Taylor series for \(\tan ^{-1} x\) would you have to use to evaluate each term on the righthand side of the equation \begin{equation} \pi=48 \tan ^{-1} \frac{1}{18}+32 \tan ^{-1} \frac{1}{57}-20 \tan ^{-1} \frac{1}{239} \end{equation} with an error of magnitude less than \(10^{-6}\) ? In contrast, the convergence of \(\sum_{n=1}^{\infty}\left(1 / n^{2}\right)\) to \(\pi^{2} / 6\) is so slow that even 50 terms will not yield two-place accuracy.

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} $$f(x)=e^{x / 3} \sin 2 x, \quad|x| \leq 2$$

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{(n !)^{n}}{n^{\left(n^{2}\right)}}$$

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

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=(n+4)^{1 /(n+4)} $$

a. Use the binomial series and the fact that \begin{equation}\frac{d}{d x} \sin ^{-1} x=\left(1-x^{2}\right)^{-1 / 2}\end{equation} to generate the first four nonzero terms of the Taylor series for \(\sin ^{-1} x .\) What is the radius of convergence? b. Series for \(\cos ^{-1} x\) Use your result in part (a) to find the first five nonzero terms of the Taylor series for \(\cos ^{-1} x .\)

Suppose that \(a_{n}>0\) and \(\lim _{n \rightarrow \infty} a_{n}=\infty .\) Prove that \(\sum a_{n}\) diverges.

Uniqueness of convergent power series a. Show that if two power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) and \(\sum_{n=0}^{\infty} b_{n} x^{n}\) are convergent and equal for all values of \(x\) in an open interval \((-c, c),\) then \(a_{n}=b_{n}\) for every \(n\) . (Hint: Let \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n} .\) Differentiate term by term to show that \(a_{n}\) and \(b_{n}\) both equal \(f^{(n)}(0) /(n !) . )\) b. Show that if \(\sum_{n=0}^{\infty} a_{n} x^{n}=0\) for all \(x\) in an open interval \((-c, c),\) then \(a_{n}=0\) for every \(n .\)

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} \frac{2^{n}-1}{3^{n}} $$

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{n^{n}}{2^{\left(n^{2}\right)}}$$

a. Series for sinh \(^{-1} x\) Find the first four nonzero terms of the Taylor series for \begin{equation} \sinh ^{-1} x=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} \end{equation} b. Use the first three terms of the series in part (a) to estimate sinh \(^{-1} 0.25 .\) Give an upper bound for the magnitude of the estimation error.

The sum of the series \(\sum_{n=0}^{\infty}\left(n^{2} / 2^{n}\right)\) . To find the sum of this series, express 1\(/(1-x)\) as a geometric series, differentiate both sides of the resulting equation with respect to \(x,\) multiply both sides of the result by \(x\) , differentiate again, multiply by \(x\) again, and set \(x\) equal to 1\(/ 2 .\) What do you get?

The limit \(L\) of an alternating series that satisfies the conditions of Theorem 15 lies between the values of any two consecutive partial sums. This suggests using the average $$\frac{s_{n}+s_{n+1}}{2}=s_{n}+\frac{1}{2}(-1)^{n+2} a_{n+1}$$ to estimate \(L .\) Compute $$s_{20}+\frac{1}{2} \cdot \frac{1}{21}$$ as an approximation to the sum of the alternating harmonic series. The exact sum is \(\ln 2=0.69314718 \ldots .\)

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{n^{n}}{\left(2^{n}\right)^{2}}$$

Suppose that \(a_{n}>0\) and \(\lim _{n \rightarrow \infty} n^{2} a_{n}=0 .\) Prove that \(\sum a_{n}\) converges.

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

Obtain the Taylor series for 1\(/(1+x)^{2}\) from the series for \(-1 /(1+x) .\)

Show that \(\sum_{n=2}^{\infty}\left((\ln n)^{q} / n^{p}\right)\) converges for \(-\infty < q <\infty\) and \(p >1.\) (Hint: Limit Comparison with \(\sum_{n=2}^{\infty} 1 / n^{r}\) for \(1 < r < p.)\)

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{i=1}^{\infty} \frac{1 \cdot 3 \cdot \cdots \cdot(2 n-1)}{4^{n} 2^{n} n !}$$

Use the Taylor series for 1\(/\left(1-x^{2}\right)\) to obtain a series for 2\(x /\left(1-x^{2}\right)^{2}\)

Show that the sum of the first 2\(n\) terms of the series $$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\cdots$$ is the same as the sum of the first \(n\) terms of the series $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\cdots$$ Do these series converge? What is the sum of the first \(2 n+1\) terms of the first series? If the series converge, what is their sum?

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot \cdots \cdot(2 n-1)}{[2 \cdot 4 \cdot \cdots \cdot(2 n)]\left(3^{n}+1\right)}$$

Decimal numbers Any real number in the interval \([0,1]\) can be represented by a decimal (not necessarily unique) as \begin{equation}0 . d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\cdots,\end{equation} where \(d_{i}\) is one of the integers \(0,1,2,3, \ldots, 9 .\) Prove that the series on the right-hand side always converges.

Neither the Ratio Test nor the Root Test helps with \(p\) -series. Try them on $$\sum_{n=1}^{\infty} \frac{1}{n^{p}}$$ and show that both tests fail to provide information about convergence.

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{2^{n}+3^{n}}{4^{n}} $$