$[-/0\mathrm{.}13$ Points$]0/100$ Submissions Used

Suppose ${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ and ${\displaystyle \underset{?}{\overset{?}{}}}{b}_n$ are series with positive terms and ${\displaystyle \underset{?}{\overset{?}{}}}{b}_n$ is known to be convergent.

$($a$)$ If $a_n>b_n$ for all $n,$ what can you say about ${\displaystyle \underset{?}{\overset{?}{}}}{a}_n?$ Why?

We cannot say anything about $a_n.$

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ converges by the Comparison Test.

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ converges if and only if $a_{n}4b_n.$

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ converges if and only if $a_{n}2b_n.$

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ diverges by the Comparison Test.

$($b$)$ If $n{\displaystyle \underset{?}{\overset{?}{}}}{a}_n{\displaystyle \underset{?}{\overset{?}{}}}{a}_n\frac{b_n}{4}{a}_n{b}_n{\displaystyle \underset{?}{\overset{?}{}}}{a}_n{\displaystyle \underset{?}{\overset{?}{}}}{a}_n\frac{b_n}{2}{a}_n{b}_n{\displaystyle \underset{?}{\overset{?}{}}}{a}_n{\displaystyle \underset{?}{\overset{?}{}}}{a}_n{a}_n$ for all $n,$ what can you say about ${\displaystyle \underset{?}{\overset{?}{}}}{a}_n?$ Why?

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ converges $if$ and only $if\frac{b_n}{4}{a}_n{b}_n.$

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ converges $by$ the Comparison Test.

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ converges $if$ and only $if\frac{b_n}{2}{a}_n{b}_n.$

${\displaystyle \underset{?}{\overset{?}{}}}{a}_n$ diverges $by$ the Comparison Test.

$We$ cannot say anything about ${\displaystyle \underset{?}{\overset{?}{}}}{a}_n.$