## Problem Set: Comparison Tests

Use the comparison test to determine whether the following series converge.

1. $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ where ${a}_{n}=\frac{2}{n\left(n+1\right)}$

2. $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ where ${a}_{n}=\frac{1}{n\left(n+\frac{1}{2}\right)}$

3. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{2\left(n+1\right)}$

4. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{2n - 1}$

5. $\displaystyle\sum _{n=2}^{\infty }\frac{1}{{\left(n\text{ln}n\right)}^{2}}$

6. $\displaystyle\sum _{n=1}^{\infty }\frac{n\text{!}}{\left(n+2\right)\text{!}}$

7. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{n\text{!}}$

8. $\displaystyle\sum _{n=1}^{\infty }\frac{\sin\left(\frac{1}{n}\right)}{n}$

9. $\displaystyle\sum _{n=1}^{\infty }\frac{{\sin}^{2}n}{{n}^{2}}$

10. $\displaystyle\sum _{n=1}^{\infty }\frac{\sin\left(\frac{1}{n}\right)}{\sqrt{n}}$

11. $\displaystyle\sum _{n=1}^{\infty }\frac{{n}^{1.2}-1}{{n}^{2.3}+1}$

12. $\displaystyle\sum _{n=1}^{\infty }\frac{\sqrt{n+1}-\sqrt{n}}{n}$

13. $\displaystyle\sum _{n=1}^{\infty }\frac{\sqrt[4]{n}}{\sqrt[3]{{n}^{4}+{n}^{2}}}$

Use the limit comparison test to determine whether each of the following series converges or diverges.

14. $\displaystyle\sum _{n=1}^{\infty }{\left(\frac{\text{ln}n}{n}\right)}^{2}$

15. $\displaystyle\sum _{n=1}^{\infty }{\left(\frac{\text{ln}n}{{n}^{0.6}}\right)}^{2}$

16. $\displaystyle\sum _{n=1}^{\infty }\frac{\text{ln}\left(1+\frac{1}{n}\right)}{n}$

17. $\displaystyle\sum _{n=1}^{\infty }\text{ln}\left(1+\frac{1}{{n}^{2}}\right)$

18. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{4}^{n}-{3}^{n}}$

19. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{2}-n\sin{n}}$

20. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{e}^{\left(1.1\right)n}-{3}^{n}}$

21. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{e}^{\left(1.01\right)n}-{3}^{n}}$

22. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{1+\frac{1}{n}}}$

23. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{{2}^{1+\frac{1}{n}}{n}^{1+\frac{1}{n}}}$

24. $\displaystyle\sum _{n=1}^{\infty }\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)$

25. $\displaystyle\sum _{n=1}^{\infty }\left(1-\cos\left(\frac{1}{n}\right)\right)$

26. $\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}\left(\frac{\pi }{2}-{\tan}^{-1}n\right)$

27. $\displaystyle\sum _{n=1}^{\infty }{\left(1-\frac{1}{n}\right)}^{n.n}$ (Hint: ${\left(1-\frac{1}{n}\right)}^{n}\to \frac{1}{e}.$)

28. $\displaystyle\sum _{n=1}^{\infty }\left(1-{e}^{-\frac{1}{n}}\right)$ (Hint: $\frac{1}{e}\approx {\left(1 - \frac{1}{n}\right)}^{n}$, so $1-{e}^{-\frac{1}{n}}\approx \frac{1}{n}.$)

29. Does $\displaystyle\sum _{n=2}^{\infty }\frac{1}{{\left(\text{ln}n\right)}^{p}}$ converge if $p$ is large enough? If so, for which $p\text{?}$

30. Does $\displaystyle\sum _{n=1}^{\infty }{\left(\frac{\left(\text{ln}n\right)}{n}\right)}^{p}$ converge if $p$ is large enough? If so, for which $p\text{?}$

31. For which $p$ does the series $\displaystyle\sum _{n=1}^{\infty }\frac{{2}^{pn}}{{3}^{n}}$ converge?

32. For which $p>0$ does the series $\displaystyle\sum _{n=1}^{\infty }\frac{{n}^{p}}{{2}^{n}}$ converge?

33. For which $r>0$ does the series $\displaystyle\sum _{n=1}^{\infty }\frac{{r}^{{n}^{2}}}{{2}^{n}}$ converge?

34. For which $r>0$ does the series $\displaystyle\sum _{n=1}^{\infty }\frac{{2}^{n}}{{r}^{{n}^{2}}}$ converge?

35. Find all values of $p$ and $q$ such that $\displaystyle\sum _{n=1}^{\infty }\frac{{n}^{p}}{{\left(n\text{!}\right)}^{q}}$ converges.

36. Does $\displaystyle\sum _{n=1}^{\infty }\frac{{\sin}^{2}\left(\frac{nr}{2}\right)}{n}$ converge or diverge? Explain.

37. Explain why, for each $n$, at least one of $\left\{|\sin{n}|,|\sin\left(n+1\right)|\text{,…},|\sin{n}+6|\right\}$ is larger than $\frac{1}{2}$. Use this relation to test convergence of $\displaystyle\sum _{n=1}^{\infty }\frac{|\sin{n}|}{\sqrt{n}}$.

38. Suppose that ${a}_{n}\ge 0$ and ${b}_{n}\ge 0$ and that $\displaystyle\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ and $\displaystyle\sum _{n=1}^{\infty }{b}^{2}{}_{n}$ converge. Prove that $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{b}_{n}$ converges and $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{b}_{n}\le \frac{1}{2}\left(\displaystyle\sum _{n=1}^{\infty }{a}_{n}^{2}+\displaystyle\sum _{n=1}^{\infty }{b}_{n}^{2}\right)$.

39. Does $\displaystyle\sum _{n=1}^{\infty }{2}^{\text{-}\text{ln}\text{ln}n}$ converge? (Hint: Write ${2}^{\text{ln}\text{ln}n}$ as a power of $\text{ln}n.$)

40. Does $\displaystyle\sum _{n=1}^{\infty }{\left(\text{ln}n\right)}^{\text{-}\text{ln}n}$ converge? (Hint: Use $n={e}^{\text{ln}\left(n\right)}$ to compare to a $p-\text{series}\text{.}$)

41. Does $\displaystyle\sum _{n=2}^{\infty }{\left(\text{ln}n\right)}^{\text{-}\text{ln}\text{ln}n}$ converge? (Hint: Compare ${a}_{n}$ to $\frac{1}{n}.$)

42. Show that if ${a}_{n}\ge 0$ and $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges, then $\displaystyle\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges. If $\displaystyle\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges, does $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ necessarily converge?

43. Suppose that ${a}_{n}>0$ for all $n$ and that $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges. Suppose that ${b}_{n}$ is an arbitrary sequence of zeros and ones. Does $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{b}_{n}$ necessarily converge?

44. Suppose that ${a}_{n}>0$ for all $n$ and that $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ diverges. Suppose that ${b}_{n}$ is an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{b}_{n}$ necessarily diverge?

45. Complete the details of the following argument: If $\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}$ converges to a finite sum $s$, then $\frac{1}{2}s=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\text{\cdots }$ and $s-\frac{1}{2}s=1+\frac{1}{3}+\frac{1}{5}+\text{\cdots }$. Why does this lead to a contradiction?

46. Show that if ${a}_{n}\ge 0$ and $\displaystyle\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges, then $\displaystyle\sum _{n=1}^{\infty }{\sin}^{2}\left({a}_{n}\right)$ converges.

47. Suppose that $\frac{{a}_{n}}{{b}_{n}}\to 0$ in the comparison test, where ${a}_{n}\ge 0$ and ${b}_{n}\ge 0$. Prove that if $\displaystyle\sum {b}_{n}$ converges, then $\displaystyle\sum {a}_{n}$ converges.

48. Let ${b}_{n}$ be an infinite sequence of zeros and ones. What is the largest possible value of $x=\displaystyle\sum _{n=1}^{\infty }\frac{{b}_{n}}{{2}^{n}}\text{?}$

49. Let ${d}_{n}$ be an infinite sequence of digits, meaning ${d}_{n}$ takes values in $\left\{0,1\text{,\ldots },9\right\}$. What is the largest possible value of $x=\displaystyle\sum _{n=1}^{\infty }\frac{{d}_{n}}{{10}^{n}}$ that converges?

50. Explain why, if $x>\frac{1}{2}$, then $x$ cannot be written $x=\displaystyle\sum _{n=2}^{\infty }\frac{{b}_{n}}{{2}^{n}}\left({b}_{n}=0\text{ or }1,{b}_{1}=0\right)$.

51. [T] Evelyn has a perfect balancing scale, an unlimited number of $1\text{-kg}$ weights, and one each of $\frac{1}{2}\text{-kg},\frac{1}{4}\text{-kg},\frac{1}{8}\text{-kg}$, and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?

52. [T] Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited collection of $1\text{-kg}$ weights, and nine each of $0.1\text{-kg,}$ $0.01\text{-kg},0.001\text{-kg,}$ and so on weights. Assuming the scale is big enough, can he do this? What does this have to do with infinite series?

53. The series $\displaystyle\sum _{n=1}^{\infty }\frac{1}{2n}$ is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which $n$ is odd. Let $m>1$ be fixed. Show, more generally, that deleting all terms $\frac{1}{n}$ where $n=mk$ for some integer $k$ also results in a divergent series.

54. In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from $\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}$ by removing any term $\frac{1}{n}$ if a given digit, say $9$, appears in the decimal expansion of $n$. Argue that this depleted harmonic series converges by answering the following questions.

1. How many whole numbers $n$ have $d$ digits?
2. How many $d\text{-digit}$ whole numbers $h\left(d\right)$. do not contain $9$ as one or more of their digits?
3. What is the smallest $d\text{-digit}$ number $m\left(d\right)\text{?}$
4. Explain why the deleted harmonic series is bounded by $\displaystyle\sum _{d=1}^{\infty }\frac{h\left(d\right)}{m\left(d\right)}$.
5. Show that $\displaystyle\sum _{d=1}^{\infty }\frac{h\left(d\right)}{m\left(d\right)}$ converges.

55. Suppose that a sequence of numbers ${a}_{n}>0$ has the property that ${a}_{1}=1$ and ${a}_{n+1}=\frac{1}{n+1}{S}_{n}$, where ${S}_{n}={a}_{1}+\cdots +{a}_{n}$. Can you determine whether $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges? (Hint: ${S}_{n}$ is monotone.)

56. Suppose that a sequence of numbers ${a}_{n}>0$ has the property that ${a}_{1}=1$ and ${a}_{n+1}=\frac{1}{{\left(n+1\right)}^{2}}{S}_{n}$, where ${S}_{n}={a}_{1}+\text{\cdots }+{a}_{n}$. Can you determine whether $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges? (Hint: ${S}_{2}={a}_{2}+{a}_{1}={a}_{2}+{S}_{1}={a}_{2}+1=1+\frac{1}{4}=\left(1+\frac{1}{4}\right){S}_{1}$, ${S}_{3}=\frac{1}{{3}^{2}}{S}_{2}+{S}_{2}=\left(1+\frac{1}{9}\right){S}_{2}=\left(1+\frac{1}{9}\right)\left(1+\frac{1}{4}\right){S}_{1}$, etc. Look at $\text{ln}\left({S}_{n}\right)$, and use $\text{ln}\left(1+t\right)\le t$, $t>0.$)