1. True or False The series $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k}\cos ( k\pi )$ is an alternating series.

2. True or False $\sum\limits_{k\,=\,1}^{\infty }[ 1+(-1) ^{k}]$ is an alternating series.

3. True or False In an alternating series, $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k}\,a_{k}$, if $\lim\limits_{n\, \rightarrow \,\infty }\,a_{n}=0,$ then the series is convergent.

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4. True or False If the alternating series $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}\,a_{k}$ satisfies the two conditions of the Alternating Series Test, then the error $E_{n}$ in using the sum $S_{n}$ of the first $n$ terms as an approximation of the sum $S$ of the series is $\vert E_{n}\vert \leq a_{n}$.

5. True or False A series that is not absolutely convergent is divergent.

6. True or False If a series is absolutely convergent, then the series converges.

7. $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}\dfrac{1}{k^{2}}$

8. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{1}{2\sqrt{k}}$

9. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k}{2k+1}$

10. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k+1}{k}$

11. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k^{2}}{5k^{2}+2}$

12. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k+1}{k^{2}}$

13. $\sum\limits_{k\,=\,1}^{\infty }\dfrac{(-1)^{k+1}}{(k+1)2^{k}}$

14. $\sum\limits_{k\,=\,2}^{\infty}(-1)^{k}\dfrac{1}{k\,\ln k}$

15. $\sum\limits_{k\,=\,2}^{\infty}(-1)^{k}\dfrac{1}{1+2^{-k}}$

16. $\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{1}{k!}$

17. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\left(\dfrac{k}{k+1}\right)^{k}$

18. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{k^{2}}{(k+1) ^{3}}$

19. $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}\dfrac{1}{k^{2}}$

20. $\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{1}{k!}$

21. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{1}{k^{4}}$

22. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\left(\dfrac{1}{\sqrt{k}}\right)^{k}$

23. $\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{1}{k!}\left( \dfrac{1}{3}\right) ^{k}$

24. $\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{1}{k!}\left( \dfrac{1}{2}\right) ^{k}$

25. $\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{1}{2k+1}\left( \dfrac{1}{3}\right) ^{2k+1}$

26. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{1}{k^{k}}$

27. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{2k}$

28. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{3k-4}$

29. $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}\dfrac{\sin k}{k^{2}+1}$

30. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{\cos k}{k^{2}}$

31. $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}\left(\dfrac{1}{5}\right)^{k}$

32. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{5^{k}}{k!}$

33. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{e^{k}}{k}$

34. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}\,2^{k}}{k^{2}}$

35. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k(k+1)}$

36. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k\sqrt{k+3}}$

37. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}\sqrt{k}}{k^{2}+1}$

38. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}\sqrt{k}}{k+1}$

39. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}~\dfrac{\ln k}{k}$

40. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}\ln k}{k^{3}}$

41. $\sum\limits_{k\,=\,1}^{\infty}(-1)^{k+1}\dfrac{1}{k\,e^{k}}$

42. $\sum\limits_{k\,=\,1}^{\infty }\dfrac{(-1)^{k+1}}{e^{k}}$

43. $\sum\limits_{k\,=\,2}^{\infty}\dfrac{(-1)^{k}}{k\,\ln k}$

44. $\sum\limits_{k\,=\,2}^{\infty}\dfrac{(-1)^{k}}{k\,(\ln k) ^{2}}$

45. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{\sin k }{k^{2}}$

46. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}\tan ^{-1}k}{k}$

47. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k^{1/k}}$

48. $\sum\limits_{k\,=\,1}^{\infty }(-1) ^{k+1}\left( \dfrac{k}{k+1}\right) ^{k}$

49. $1-\dfrac{1}{2!}+\dfrac{1}{3!}-\dfrac{1}{4!}+\dfrac{1}{5!}-\cdots$

50. $1-\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}-\dfrac{1}{7^{2}}+\cdots$

51. Show that the positive terms of $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k}$ diverge.

52. Show that the negative terms of $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k}$ diverge.

53. Show that the terms of the series $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k}$ can be rearranged so the resulting series converges to $0$.

54. Show that the terms of the series $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k}$ can be rearranged so the resulting series converges to $2$.

55. Show that the terms of the series $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(-1)^{k+1}}{k}$ can be rearranged so the resulting series diverges.

56. Show that the series $$e^{-x}\cos x+e^{-2x}\cos (2x) +e^{-3x}\cos (3x)+\cdots$$

    is absolutely convergent for all positive values of $x$. (Hint: Use the fact that $\vert \cos \theta \vert \leq 1$.)

57. Determine whether the series below converges (absolutely or conditionally) or diverges. $$1+r\cos \theta +r^{2}\cos\! \left( 2\theta \right) +r^{3}\cos (3\theta) +\cdots$$

58. What is wrong with the following argument? $$\begin{eqnarray*} A &=& 1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{7}-\dfrac{1}{8}+\cdots \\[3pt] \left(\dfrac{1}{2}\right) A &=& \dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{6}-\dfrac{1}{8}+\cdots \end{eqnarray*}$$ So, $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A+\left(\dfrac{1}{2}\right) A = 1+\dfrac{1}{3}-\dfrac{1}{2}+\dfrac{1}{5}+\dfrac{1}{7}-\dfrac{1}{4}+\cdots$$

    590

    The series on the right is a rearrangement of the terms of the series $A$. So its sum is $A$, meaning $$\begin{array}{l} A + \left( {\frac{1}{2}} \right)A &=& A \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, A &=& 0 \\ \end{array}$$

    But, $$A = \left( {1 - \frac{1}{2}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + \left( {\frac{1}{5} - \frac{1}{6}} \right) + \cdots > 0$$

    Bernoulli’s Error In Problems 59–61, consider an incorrect argument given by Jakob Bernoulli to prove that $$\sum\limits_{k = 1}^\infty {\frac{1}{{k(k + 1)}} = \frac{1}{2} + \frac{1}{6} + \frac{1}{{12}} + \cdots = 1}$$

    Bernoulli’s argument went as follows: $$Let \,N = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots ., Then$$ $$N - 1 =\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots .$$

    Now, subtract term-by-term to get $$\begin{array}{l} N - (N - 1) = \left( {1 - \frac{1}{2}} \right) + \left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + \cdots \,\,\,\,\,(4) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 = \frac{1}{2} + \frac{1}{6} + \frac{1}{{12}} + \cdots \\ \end{array}$$

59. What is wrong with Bernoulli’s argument?

60. In general, what can be said about the convergence or divergence of a series formed by taking the term-by-term difference (or sum) of two divergent series? Support your answer with examples.

61. Although the method is wrong. Bernoulli’s conclusion is correct; that is, it is true that $\sum\limits_{k=1}^{\infty }\dfrac{1}{k(k+1) }=1.$ Prove it! [Hint: Look at the partial sums using the form of the series in (4).]

62. Show that if two series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ and $\sum\limits_{k\,=\,1}^{\infty }b_{k}$ are absolutely convergent, the series $\sum\limits_{k\,=\,1}^{\infty }(a_{k}+b_{k})$ is also absolutely convergent. Moreover, $\sum\limits_{k\,=\,1}^{\infty}\,(a_{k}+b_{k})=\sum\limits_{k\,=\,1}^{\infty}a_{k}+\sum\limits_{k\,=\,1}^{\infty }b_{k}$.

63. Show that if a series converges absolutely, then the series consisting of just the positive terms converges, as does the series consisting of just the negative terms.

64. Show that if a series converges conditionally, then the series consisting of just the positive terms diverges, as does the series consisting of just the negative terms.

65. Determine whether the series $\sum\limits_{k=1}^{\infty }c_{k}$, where $$\,\,{c_k} = \left\{ \begin{array}{l} \frac{1}{{{a^k}}}\,\,\,\,\qquad{\rm{if}}\,k\,{\rm{is}}\,{\rm{even}} \\ - \frac{1}{{{b^k}}}\qquad{\rm{if}}\,k\,{\rm{is}}\,{\rm{odd}} \\ \end{array} \right.a > 1,\,b > 1,$$

    converges absolutely, converges conditionally, or diverges.

66. Prove that if a series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ is absolutely convergent, any rearrangement of its terms results in a series that is also absolutely convergent. (Hint: Use the triangle inequality: $\left\vert a+b\right\vert \leq \vert a \vert+\left\vert b\right\vert$.)

67. Suppose that the terms of the series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ are alternating, $\vert a_{k+1}\vert \leq \left\vert a_{k}\right\vert$ for all integers $k,$ and $\lim\limits_{n\rightarrow \infty }a_{n}=0.$ Show that the series converges.

68. $\sum\limits_{k=1}^{\infty }\sin \dfrac{(-1)^{k}}{k}$

69. $\sum\limits_{k=2}^{\infty }\dfrac{(-1)^{k}}{\sqrt[p]{k^{3}+1}},~p>2$

70. $\sum\limits_{k=2}^{\infty }(-1)^{k}k^{(1-k)/k}$

71. $\sum\limits_{k=2}^{\infty }\dfrac{(-1)^{k}}{(\ln k) ^{\ln k}}$

72. $\sum\limits_{k\,=\,1}^{\infty}(-1) ^{k+1}\dfrac{\sqrt{k}}{(k+1) }$

73. $\sum\limits_{k\,=\,1}^{\infty}(-1) ^{k+1}\dfrac{k}{(k+1) ^{2}}$

74. $\sum\limits_{k\,=\,2}^{\infty}(-1) ^{k}\ln \dfrac{k+1}{k}$

75. Let $\{a_{n}\}$ be a sequence that is decreasing and is bounded from below by $0$. Define $$R_{n}=\sum_{k=n+1}^{\infty }(-1)^{k+1}a_{k} \qquad \hbox{and}\qquad \Delta a_{k}=a_{k}-a_{k+1}$$

    Suppose that the sequence $\left\{ \Delta a_{k}\right\}$ decreases.

    1. (a) Show that the series $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k+1}~\Delta a_{k}$ is a convergent alternating series.
    2. (b) Show that $$\left\vert R_{n}\right\vert =\frac{a_{n}}{2}+\frac{1}{2}\sum_{k\,=\,1}^{\infty }(-1)^{k}~\Delta \,a_{k+n-1}$$ and $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k}~\Delta \,a_{k+n-1}<0$. Deduce $\left\vert R_{n}\right\vert <\dfrac{1}{2}\,a_{n}$.
    3. (c) Show that $$\left\vert R_{n}\right\vert =\frac{a_{n+1}}{2}+\frac{1}{2} \sum_{k\,=\,1}^{\infty }(-1)^{k+1}~\Delta \,a_{k+n}$$
    4. (d) Conclude that $\dfrac{a_{n+1}}{2}<\left\vert R_{n}\right\vert$.

Source: Based on R. Johnsonbaugh, Summing an alternating series, American Mathematical Monthly, Vol. 86, No. 8 (Oct. 1979), pp. 637-648.