Remainders in alternating series Determine how

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\sum\left|a_{k}\right|\) diverges. e. If \(\sum_{ }^{ }a_k^{\ 2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\sum_{ }^{ }a_k^{\ 2}\) converges. g. If \(\sum a_{k}\) converges conditionally, then \(\sum\left|a_{k}\right|\) diverges.

Describe how to apply the Alternating Series Test.

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3}}\)

Is it possible for a series of positive terms to converge conditionally? Explain.

Why does absolute convergence imply convergence?

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3 / 2}}\)

35-38. Estimating infinite sums Estimate the value of the following convergent series with an absolute error less than \(10^{-3}\). \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{(2 k+1)^{3}}\)

Suppose an alternating series converges to a value L. Explain how to estimate the remainder that occurs when the series is terminated after n terms.

35-38. Estimating infinite sums Estimate the value of the following convergent series with an absolute error less than \(10^{-3}\). \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{5}}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=0}^{\infty} \frac{(-1)^{k}}{\sqrt{k^{2}+4}}\)

A better remainder Suppose an alternating series \(\sum_{k=1}^{\infty}(-1)^k\ a_k\) converges to S and the sum of the first n terms of the series is \(S_{n}\). Suppose also that the difference between the magnitudes of consecutive terms decreases with k. Then it can be shown that for \(n \geq 1\) \(\left|S-\left(S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2}\right)\right| \leq \frac{1}{2}\left|a_{n+1}-a_{n+2}\right|\) a. Interpret this inequality and explain why it gives a better approximation to S than simply using \(S_{n}\) to approximate S. b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than \(10^{-6}\) using both \(S_{n}\) and the method explained in part (a). (i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\)

A fallacy Explain the fallacy in the following argument. Let \(x=\frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\) and \(y=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots\). It follows that 2y = x + y, which implies that x = y. On the other hand, \(x-y=(\underbrace{\left.1-\frac{1}{2}\right)}_{>0}+(\underbrace{\left.\frac{1}{3}-\frac{1}{4}\right)}_{>0}+(\underbrace{\frac{1}{5}-\frac{1}{6}}_{>0})+\cdots>0\) is a sum of positive terms, so x > y. Thus, we have shown that x = y and x > y.

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty}\frac{(-1)^k\ k^2}{\sqrt{k^6+1}}\)

Alternating p-series Given that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\), show that \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{2}}=\frac{\pi^{2}}{12}\). (Assume the result of Exercise 53.)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{\pi^{3}}{32}=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k+1)^{3}}\)

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty} \frac{(-1)^{k} \tan ^{-1} k}{k^{3}}\)

Alternating Series Test Show that the series \(\frac{1}{3}-\frac{2}{5}+\frac{3}{7}-\frac{4}{9}+\cdots=\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{2 k+1}\) diverges. Which condition of the Alternating Series Test is not satisfied?

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=2}^{\infty}(-1)^{k}\left(1+\frac{1}{k}\right)\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=0}^{\infty}\left(-\frac{1}{5}\right)^{k}\)

Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the conditionally convergent) alternating harmonic series has the value \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots=\ln 2\). Show that by rearranging the terms (so the sign pattern is + + -), \(1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\cdots=\frac{3}{2} \ln 2\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{10}+2 k^{5}+1}{k\left(k^{10}+1\right)}\)

Explain why the sequence of partial sums for an alternating series is not an increasing sequence.

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{\pi^{2}}{12}=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{2}}\)

Why does the value of a converging alternating series lie between any two consecutive terms of its sequence of partial sums?

Alternating p-series Given that \(\sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}\), show that \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}}=\frac{7 \pi^{4}}{720}\). (Assume the result of Exercise 53.)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{2}}{k^{3}+1}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=2}^{\infty}(-1)^{k} \frac{k^{2}-1}{k^{2}+3}\)

Geometric series In Section 8.3, we established that the geometric series \(\sum r^{k}\) converges provided |r| < 1. Notice that if -1 < r < 0, the geometric series is also an alternating series. Use the Alternating Series Test to show that for -1 < r < 0, the series \(\sum r^{k}\) converges.

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln ^{2} k}\)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{1}{e}=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty}(-1)^k\ k\sin\left(\frac{1}{k}\right)\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k^{2}+10}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty} \frac{\cos \pi k}{k^{2}}\)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{\pi \sqrt{3}}{9}-\frac{\ln 2}{3}=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{3 k+2}\)

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}\)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{\pi}{4}=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2 k+1}\)

Give an example of a convergent alternating series that fails to converge absolutely.

35-38. Estimating infinite sums Estimate the value of the following convergent series with an absolute error less than \(10^{-3}\). \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2 k+1) !}\)

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty}\frac{(-1)^k\ k}{2k+1}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=2}^{\infty}(-1)^{k} \frac{\ln k}{k^{2}}\)

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1} e^{k}}{(k+1) !}\)

Is it possible for an alternating series to converge absolutely, but not conditionally?

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}\)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{\pi \sqrt{3}}{9}+\frac{\ln 2}{3}=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{3 k+1}\)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\ln 2=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k !}{k^{k}}\)

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally. \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\ln k}\)

35-38. Estimating infinite sums Estimate the value of the following convergent series with an absolute error less than \(10^{-3}\). \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{k}}\)

Give an example of a series that converges conditionally but not absolutely

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{7 \pi^{4}}{720}=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}}\)

11-24. Alternating Series Test Determine whether the following series converge. \(\sum_{k=1}^{\infty}(-1)^{k+1} k^{1 / k}\)

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\frac{31 \pi^{6}}{30,240}=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{6}}\)

Explain why the remainder in terminating an alternating series is less than the first neglected term.

25-34. Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4}\). Although you do not need it, the exact value of the series is given in each case. \(\pi=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{4^{k}}\left(\frac{2}{4 k+1}+\frac{2}{4 k+2}+\frac{1}{4 k+3}\right)\)

Remainders in alternating series Given any infinite series \(\sum a_{k}\), let N(r) be the number of terms of the series that must be summed to guarantee that the remainder is less than \(10^{-r}\), where r is a positive integer. a. Graph the function N(r) for the three alternating p-series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{p}}\), for p = 1, 2, and 3. Compare the three graphs and discuss what they mean about the rates of convergence of the three series. b. Carry out the procedure of part (a) for the series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}\) and compare the rates of convergence of all four series.