Chapter 10

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

If \(\sum a_{n}\) is a convergent series of positive terms, prove that \(\sum \sin \left(a_{n}\right)\) converges.

Show that if \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely, then $$\left|\sum_{n=1}^{\infty} a_{n}\right| \leq \sum_{n=1}^{\infty}\left|a_{n}\right|$$

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

Show that neither the Ratio Test nor the Root Test provides information about the convergence of $$\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{p}} \quad(p constant )$$

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

Series for sin \(^{-1} x\) Integrate the binomial series for \(\left(1-x^{2}\right)^{-1 / 2}\) to show that for \(|x|<1\) \begin{equation} \sin ^{-1} x=x+\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)} \frac{x^{2 n+1}}{2 n+1} \end{equation}

Show that if \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) both converge absolutely, then so do the following. $$\begin{array}{ll}{\text { a. }} & {\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)} & {\text { b. } \sum_{n=1}^{\infty}\left(a_{n}-b_{n}\right)} \\ {\text { c. }} & {\sum_{n=1}^{\infty} k a_{n}(k \text { any number })}\end{array}$$

Let $$a_{n}=\left\\{\begin{array}{ll}{n / 2^{n},} & {\text { if } n \text { is a prime number }} \\ {1 / 2^{n},} & {\text { otherwise. }}\end{array}\right.$$ Does \(\sum a_{n}\) converge? Give reasons for your answer.

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

Series for tan \(^{-1} x\) for \(|x|>1\) Derive the series \begin{equation} \tan ^{-1} x=\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3 x^{3}}-\frac{1}{5 x^{5}}+\cdots, \quad x>1 \end{equation} \begin{equation} \tan ^{-1} x=-\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3 x^{3}}-\frac{1}{5 x^{5}}+\cdots, \quad x<-1 \end{equation} by integrating the series \begin{equation} \frac{1}{1+t^{2}}=\frac{1}{t^{2}} \cdot \frac{1}{1+\left(1 / t^{2}\right)}=\frac{1}{t^{2}}-\frac{1}{t^{4}}+\frac{1}{t^{6}}-\frac{1}{t^{8}}+\cdots \end{equation} in the first case from \(x\) to \(\infty\) and in the second case from \(-\infty\) to \(x\)

Show by example that \(\sum_{n=1}^{\infty} a_{n} b_{n}\) may diverge even if \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) both converge.

Show that \(\sum_{n=1}^{\infty} 2^{\left(n^{2}\right)} / n !\) diverges. Recall from the Laws of Exponents that \(2^{\left(n^{2}\right)}=\left(2^{n}\right)^{n}\)

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

If \(\sum a_{n}\) converges absolutely, prove that \(\sum a_{n}^{2}\) converges.

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

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

Does the series $$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right)$$ converge or diverge? Justify your answer.

In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to \(-1 / 2 .\) Start the new arrangement with the first negative term, which is \(-1 / 2 .\) Whenever you have a sum that is less than or equal to \(-1 / 2,\) start introducing positive terms, taken in order, until the new total is greater than \(-1 / 2 .\) Then add negative terms until the total is less than or equal to \(-1 / 2\) again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If \(s_{n}\) is the sum of the first \(n\) terms of your new series, plot the points \(\left(n, s_{n}\right)\) to illustrate how the sums are behaving.

In each of the geometric series in Exercises \(69-72,\) write out the first few terms of the series to find \(a\) and \(r,\) and find the sum of the series. Then express the inequality \(|r| < 1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$ \sum_{n=0}^{\infty}(-1)^{n} x^{n} $$

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

Show that \begin{equation} \text { a. }\cosh i \theta=\cos \theta, \quad \text { b. } \sinh i \theta=i \sin \theta \end{equation}

In each of the geometric series in Exercises \(69-72,\) write out the first few terms of the series to find \(a\) and \(r,\) and find the sum of the series. Then express the inequality \(|r| < 1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$ \sum_{n=0}^{\infty}(-1)^{n} x^{2 n} $$

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

Outline of the proof of the Rearrangement Theorem (Theorem 17\()\) $$\begin{array}{l}{\text { a. Let } \epsilon \text { be a positive real number, let } L=\sum_{n=1}^{\infty} a_{n}, \text { and let }} \\\ {s_{k}=\sum_{n=1}^{k} a_{n} \text { . Show that for some index } N_{1} \text { and for some }} \\ {\quad \text { index } N_{2} \geq N_{1}}\end{array}$$ $$\sum_{n=N_{1}}^{\infty}\left|a_{n}\right|<\frac{\epsilon}{2} \quad \text { and } \quad\left|s_{N_{2}}-L\right|<\frac{\epsilon}{2}$$ $$\begin{array}{l}{\text { Since all the terms } a_{1}, a_{2}, \ldots, a_{N_{2}} \text { appear somewhere in the }} \\ {\text { sequence }\left\\{b_{n}\right\\}, \text { there is an index } N_{3} \geq N_{2} \text { such that if }} \\ {n \geq N_{3}, \text { then }\left(\sum_{k=1}^{n} b_{k}\right)-s_{N_{2}} \text { is at most a sum of terms } a_{m}} \\ {\text { with } m \geq N_{1} . \text { Therefore, if } n \geq N_{3}}\end{array}$$ $$\begin{aligned}\left|\sum_{k=1}^{n} b_{k}-L\right| & \leq\left|\sum_{k=1}^{n} b_{k}-s_{N_{2}}\right|+\left|s_{N_{2}}-L\right| \\ & \leq \sum_{k=N_{1}}^{\infty}\left|a_{k}\right|+\left|s_{N_{2}}-L\right|<\epsilon \end{aligned}$$ $$\begin{array}{l}{\text { b. The argument in part (a) shows that if } \sum_{n=1}^{\infty} a_{n} \text { converges }} \\ {\text { absolutely then } \sum_{n=1}^{\infty} b_{n} \text { converges and } \sum_{n=1}^{\infty} b_{n}=\sum_{n=1}^{\infty} a_{n}} \\ {\text { Now show that because } \sum_{n=1}^{\infty}\left|a_{n}\right| \text { converges, } \sum_{n=1}^{\infty}\left|b_{n}\right|} \\ {\text { converges to } \sum_{n=1}^{\infty}\left|a_{n}\right|}\end{array}$$

By multiplying the Taylor series for \(e^{x}\) and \(\sin x,\) find the terms through \(x^{5}\) of the Taylor series for \(e^{x} \sin x .\) This series is the imaginary part of the series for \begin{equation} e^{x} \cdot e^{i x}=e^{(1+i) x} \end{equation} Use this fact to check your answer. For what values of \(x\) should the series for \(e^{x}\) sin \(x\) converge?

It is not yet known whether the series \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n}\end{equation} converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. \begin{equation} \begin{array}{l}{\text { a. Define the sequence of partial sums }}\end{array} \end{equation} \begin{equation} s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n}. \end{equation} \begin{equation} \begin{array}{l}{\text { What happens when you try to find the limit of } s_{k} \text { as } k \rightarrow \infty ?} \\ {\text { Does your CAS find a closed form answer for this limit? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Plot the first } 100 \text { points }\left(k, s_{k}\right) \text { for the sequence of partial }} \\ \quad {\text { sums. Do they appear to converge? What would you estimate }} \\ \quad {\text { the limit to be? }} \\ {\text { c. Next plot the first } 200 \text { points }\left(k, s_{k}\right) . \text { Discuss the behavior in }} \\ \quad {\text { your own words. }} \\ {\text { d. Plot the first } 400 \text { points }\left(k, s_{k}\right) \text { . What happens when } k=355 \text { ? }} \\\ \quad {\text { Calculate the number } 355 / 113 . \text { Explain from you calculation }} \\ \quad {\text { what happened at } k=355 . \text { For what values of } k \text { would you }} \\ \quad {\text { guess this behavior might occur again? }} \end{array} \end{equation}

In each of the geometric series in Exercises \(69-72,\) write out the first few terms of the series to find \(a\) and \(r,\) and find the sum of the series. Then express the inequality \(|r| < 1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$ \sum_{n=0}^{\infty} 3\left(\frac{x-1}{2}\right)^{n} $$

When \(a\) and \(b\) are real, we define \(e^{(a+i b) x}\) with the equation \begin{equation} e^{(a+i b) x}=e^{a x} \cdot e^{i b x}=e^{a x}(\cos b x+i \sin b x) \end{equation} Differentiate the right-hand side of this equation to show that \begin{equation} \frac{d}{d x} e^{(a+i b) x}=(a+i b) e^{(a+i b) x} \end{equation} Thus the familiar rule \((d / d x) e^{k x}=k e^{k x}\) holds for \(k\) complex as well as real.

In each of the geometric series in Exercises \(69-72,\) write out the first few terms of the series to find \(a\) and \(r,\) and find the sum of the series. Then express the inequality \(|r| < 1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2}\left(\frac{1}{3+\sin x}\right)^{n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(1-\frac{1}{n^{2}}\right)^{n} $$

Use the definition of \(e^{i \theta}\) to show that for any real numbers \(\theta, \theta_{1}\) ,and \(\theta_{2}\) , \begin{equation} \text { a. } e^{i \theta_{1}} e^{i \theta_{2}}=e^{i\left(\theta_{1}+\theta_{2}\right)}, \quad \text { b. } e^{-i \theta}=1 / e^{i \theta} \end{equation}

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty} 2^{n} x^{n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{3^{n} \cdot 6^{n}}{2^{-n} \cdot n !} $$

Two complex numbers \(a+i b\) and \(c+i d\) are equal if and only if \(a=c\) and \(b=d .\) Use this fact to evaluate \begin{equation} \int e^{a x} \cos b x d x \quad \text { and } \int e^{a x} \sin b x d x \end{equation} from \begin{equation} \int e^{(a+i b) x} d x=\frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x}+C \end{equation} where \(C=C_{1}+i C_{2}\) is a complex constant of integration.

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty}(-1)^{n} x^{-2 n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(10 / 11)^{n}}{(9 / 10)^{n}+(11 / 12)^{n}} $$

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty}(-1)^{n}(x+1)^{n} $$

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^{n}(x-3)^{n} $$

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty} \sin ^{n} x $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{n^{2}}{2 n-1} \sin \frac{1}{n} $$

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty}(\ln x)^{n} $$

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

Make up an infinite series of nonzero terms whose sum is a. 1\(\quad\) b. \(-3 \quad\) c. 0

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

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

Show by example that \(\sum\left(a_{n} / b_{n}\right)\) may diverge even though \(\Sigma a_{n}\) and \(\sum b_{n}\) converge and no \(b_{n}\) equals \(0 .\)

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(\frac{1}{3}\right)^{n}+\frac{1}{\sqrt{2^{n}}} $$

Show by example that \(\sum\left(a_{n} / b_{n}\right)\) may converge to something other than \(A / B\) even when \(A=\Sigma a_{n}, B=\Sigma b_{n} \neq 0,\) and no \(b_{n}\) equals \(0 .\)

If \(\sum a_{n}\) converges and \(a_{n}>0\) for all \(n,\) can anything be said about \(\sum\left(1 / a_{n}\right) ?\) Give reasons for your answer.