비교 판정법

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

Does $\sum_{n=2}^{\infty }{(\ln n)}^{-\ln \ln n}$ converge? (Hint: Compare ${a}_{n}$ to $1/n.)$

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

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

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

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

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

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

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

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

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

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

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

[T] Evelyn has a perfect balancing scale, an unlimited number of $1\,\text{-kg}$ weights, and one each of $1/2\,\text{-kg},\,1/4\,\text{-kg},\,1/8\,\text{-kg},$ 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?

[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?

The series $\sum_{n=1}^{\infty }\frac{1}{2n}$ 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 $1/n$ where $n=mk$ for some integer $k$ also results in a divergent series.

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 $\sum_{n=1}^{\infty }\frac{1}{n}$ by removing any term $1/n$ 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.

a. How many whole numbers $n$ have $d$ digits? b. How many $d\text{-digit}$ whole numbers $h(d).$ do not contain $9$ as one or more of their digits? c. What is the smallest $d\text{-digit}$ number $m(d)?$ d. Explain why the deleted harmonic series is bounded by $\sum_{d=1}^{\infty }\frac{h(d)}{m(d)}.$ e. Show that $\sum_{d=1}^{\infty }\frac{h(d)}{m(d)}$ converges.

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 $\sum_{n=1}^{\infty }{a}_{n}$ converges? (Hint: ${S}_{n}$ is monotone.)

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

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If $\lim_{n\to \infty }{a}_{n}=0,$ then $\sum_{n=1}^{\infty }{a}_{n}$ converges.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If $\lim_{n\to \infty }{a}_{n}\ne 0,$ then $\sum_{n=1}^{\infty }{a}_{n}$ diverges.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If $\sum_{n=1}^{\infty }|{a}_{n}|$ converges, then $\sum_{n=1}^{\infty }{a}_{n}$ converges.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If $\sum_{n=1}^{\infty }{2}^{n}{a}_{n}$ converges, then $\sum_{n=1}^{\infty }{(-2)}^{n}{a}_{n}$ converges.