1. True or False The Ratio Test can be used to show that the series $\sum\limits_{k\,=\,1}^{\infty }\cos ( k\pi )$ diverges.

2. True or False In using the Ratio Test, if $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}} \right\vert =L,$ then the sum of the series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ equals $L.$

3. True or False In using the Ratio Test, if $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}} \right\vert =1$, then the Ratio Test indicates that the series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ converges.

4. True or False The Root Test works well if the $n$th term of a series of nonzero terms involves an $n$th root.

5. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{4k^{2}-1}{2^{k}}$

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

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

8. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{5^{k}}{k^{2}}$

9. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{10^{k}}{(2k)!}$

10. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(2k)!}{5^{k}3^{k-1}}$

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

12. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{(k+1)!}{3^{k}}$

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

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

15. $\sum\limits_{k\,=\,1}^{\infty }\dfrac{k^{3}}{k!}$

16. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{k!}{k^{k+1}}$

17. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{3^{k-1}}{k\cdot 2^{k}}$

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

19. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{k}{e^{k}}$

20. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{e^{k}}{k^{3}}$

21. $\sum\limits_{k\,=\,1}^{\infty }k\cdot 2^{k}$

22. $\sum\limits_{k\,=\,1}^{\infty }\dfrac{4^{k}}{k}$

23. $\sum\limits_{k\,=\,1}^{\infty }\left( \dfrac{2k+1}{5k+1}\right) ^{\!\!k}$

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

25. $\sum\limits_{k\,=\,1}^{\infty}\left( \dfrac{k}{5}\right) ^{\!\!k}$

26. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{\pi^{2k}}{k^{k}}$

27. $\sum\limits_{k\,=\,2}^{\infty}\left(\dfrac{\ln k}{k}\right) ^{\!\!k}$

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

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

30. $\sum\limits_{k\,=\,1}^{\infty}\left(\dfrac{\sqrt{4k^{2}+1}}{k}\right) ^{\!\!k}$

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

32. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{k^{3}}{3^{k}}$

33. $\sum\limits_{k\,=\,1}^{\infty }\dfrac{k^{4}}{5^{k}}$

34. $\sum\limits_{k\,=\,1}^{\infty }\dfrac{k}{3^{k}}$

35. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{10}{(3k+1)^{k}}$

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

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

38. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{k!}{(3k+1) !}$

39. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{k\ln k}{2^{k}}$

40. $\sum\limits_{k\,=\,1}^{\infty}\left[ \ln \left( e^{3}+\dfrac{1}{k}\right)\right]^{k}$

41. $\sum\limits_{k\,=\,1}^{\infty}\sin ^{k}\left(\dfrac{1}{k}\right)$

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

43. $\sum\limits_{k\,=\,1}^{\infty}\dfrac{\left( 1+\dfrac{1}{k}\right) ^{\!\!2k}}{e^{k}}$

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

45. For the divergent series $\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k}$, show that $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert =1$.

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46. For the convergent series $\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{ k^{2}}$, show that $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{ a_{n+1}}{a_{n}}\right\vert =1$.

47. Give an example of a convergent series $\sum\limits_{k\,= \,1}^{\infty }a_{k}$ for which $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert \neq \infty$ does not exist.

48. Give an example of a divergent series $\sum\limits_{k\,= \,1}^{\infty }a_{k}$ for which $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert \neq \infty$ does not exist.

49. 1. (a) Show that the series $\sum\limits_{k=1}^{\infty }\dfrac{ (-1) ^{k}3^{k}}{k!}$ converges.
    2. (b) Use technology to find the sum of the series.

50. Determine whether the following series is convergent or divergent: $$\frac{1}{3}-\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}-\frac{4^{3}}{3^{4}} +\cdots +\frac{(-1)^{n-1}n^{3}}{3^{n}}+\cdots$$

51. Show that $\lim\limits_{n\,\rightarrow \,\infty }\,\dfrac{n!}{n^{n}}=0$, where $n$ denotes a positive integer.

52. Show that the Root Test is inconclusive for $\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k}$ and $\sum\limits_{k\,= \,1}^{\infty }\dfrac{1}{k^{2}}$.

53. Use the Ratio Test to find the real numbers $x$ for which the series $\sum\limits_{k\,=\,1}^{\infty }\dfrac{x^{k}}{k^{2}}$ converges or diverges.

54. Prove that $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ diverges if $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}} \right\vert =\infty$.

55. Prove the Root Test.

56. The terms of the series $$\dfrac{1}{4}+\dfrac{1}{2}+\dfrac{1}{8}+\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{8 }+\dfrac{1}{32}+\cdots$$

    are $a_{2k}=\dfrac{1}{2^{k}}$ and $a_{2k-1}=\dfrac{1}{2^{k+1}}.$

    1. (a) Show that using the Ratio Test to determine whether the series converges is inconclusive.
    2. (b) Show that using the Root Test to determine whether the series converges is conclusive.
    3. (c) Does the series converge?

57. Show that the following series converges: $$1+\frac{2}{2^{2}}+\frac{3}{3^{3}}+\frac{1}{4^{4}}+\frac{2}{5^{5}}+\frac{3}{ 6^{6}}+\cdots$$

58. Show that $\sum\limits_{k\,=\,1}^{\infty }\dfrac{(k+1)^{2}}{(k+2) !}$ converges.

    (Hint: Use the Limit Comparison Test and the convergent series $\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k!}.$)

59. Suppose $0<a<b<1$. Use the Root Test to show that the series $$a+b+a^{2}+b^{2}+a^{3}+b^{3}+\cdots$$

    converges.

60. Show that if the Ratio Test indicates a series converges, then so will the Root Test. The converse is not true. Refer to Problem 56.