5.4 Comparison tests  (Page 5/7)

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?

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?

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

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

Suppose that $a_n\text{/}{b}_n\to 0$ 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.

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

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

Explain why, if $x>1\text{/}2,$ then $x$ cannot be written $x={\displaystyle \sum _{n=2}^{\infty }\frac{b_n}{2^n}}\left(b_n=0\text{or}1,b_1=0\right).$

[T] Evelyn has a perfect balancing scale, an unlimited number of $1\text{-kg}$ weights, and one each of $1\text{/}2\text{-kg},1\text{/}4\text{-kg},1\text{/}8\text{-kg},$ 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 ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{2n}}$ 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 $1\text{/}n$ 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 ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$ by removing any term $1\text{/}n$ 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.

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 $S_n=a_1+\text{⋯}+a_n.$ Can you determine whether ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges? ( Hint: $S_n$ is monotone.)

Suppose that a sequence of numbers $a_n>0$ has the property that $a_1=1$ and $a_{n+1}=\frac{1}{{\left(n+1\right)}^2}{S}_n,$ where $S_n=a_1+\text{⋯}+a_n.$ Can you determine whether ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges? ( Hint: $S_2=a_2+a_1=a_2+S_1=a_2+1=1+1\text{/}4=\left(1+1\text{/}4\right)S_1,$ $S_3=\frac{1}{3^2}{S}_2+S_2=\left(1+1\text{/}9\right)S_2=\left(1+1\text{/}9\right)\left(1+1\text{/}4\right)S_1,$ etc. Look at $\text{ln}\left(S_n\right),$ and use $\text{ln}\left(1+t\right)\le t,$ $t>0.)$