91Ó°ÊÓ

Consider the statement :"if $0≤f\left(x\right)≤g\left(x\right)$for every x> 0 and the improper integral ${∫}_0^{∞}g\left(x\right)dx$converges then the improper integral ${∫}_0^{∞}f\left(x\right)$dx converges".

To determine whether the statement is true or false .

The improper integral ${∫}_0^{∞}g\left(x\right)dx$is convergent .

${∫}_0^{∞}g\left(x\right)dx=A$

Given that $0≤f\left(x\right)≤g\left(x\right)$

$0≤{∫}_0^{∞}f\left(x\right)dx≤{∫}_0^{∞}g\left(x\right)dx$

${∫}_0^{∞}f\left(x\right)dx≤A$

The improper integral ${∫}_0^{∞}f\left(x\right)dx$converges

Hence the statement is true.

consider the statement :" if $0≤f\left(x\right)≤g\left(x\right)$for every x>0 and$\underset{x→∞}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=3$ then the improper integral ${∫}_0^{∞}g\left(x\right)dxand{∫}_0^{∞}f\left(x\right)dx$both converge

To determine whether the statement is true or false

Consider $g\left(x\right)=\frac{1}{x^2}andf\left(x\right)=\frac{3}{x^2}$

The value of $\underset{x→∞}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x→∞}{\lim }3=3$

but the integrals,${∫}_0^{∞}g\left(x\right)dx={∫}_0^{∞}\frac{1}{x^2}and{∫}_0^{∞}f\left(x\right)dx={∫}_0^{∞}\frac{3}{x^2}$ diverges

Hence the statement is false

Consider the statement : " if $0≤a_k<\frac{1}{k}$for every positive integer k, then the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$converges"

To determine whether the given statement is true or false

using comparison test ,

It states that $\underset{k=1}{\overset{∞}{∑}}{a}_{k}and\underset{k=1}{\overset{∞}{∑}}{b}_k$be two series with positive terms such that $0≤a_k≤b_k$for every positive integer k.

If the series $\underset{k=1}{\overset{∞}{∑b_k}}$converges , then the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$also converges .

The series $\underset{k=1}{\overset{∞}{∑}}{b}_k=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k}$is divergent by the p-series,

The series $\underset{k=1}{\overset{∞}{∑}}{a}_k$is divergent.

Hence the statement is false .

Consider the statement "if $\frac{1}{k^2}<b_k$for every positive integer k ,then the series $\underset{k=1}{\overset{∞}{∑}}{b}_k$diverges .

To determine whether the given statement is true or false.

using comparison test

It states that $\underset{k=1}{\overset{∞}{∑}}{a}_{k}and\underset{k=1}{\overset{∞}{∑}}{b}_k$be two series with positive terms such that $0≤a_k≤b_k$for every positive integer k . if the series $\underset{k=1}{\overset{∞}{∑}}{b}_k$converges then The series $\underset{k=1}{\overset{∞}{∑}}{a}_k$also converges.

The test fails to determine the converges and diverges of the series $\underset{k=1}{\overset{∞}{∑}}{b}_k$.

We cannot be said the behavior of $\underset{k=1}{\overset{∞}{∑}}{b}_k$if $\frac{1}{k^2}<b_k$holds

Hence the statement is false .

Consider the statement if $a_k≤b_k$for every positive integer k and the series $\underset{k=1}{\overset{∞}{∑}}{b}_k$converges , then the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$converges

To determine whether the statement is true or false.

Consider the series $\underset{k=1}{\overset{∞}{∑}}{b}_k=\frac{1}{k^2}and\underset{k=1}{\overset{∞}{∑}}{a}_k=-\frac{1}{k}$

Clearly $a_k≤b_k$holds as:

$-\frac{1}{k}<\frac{1}{k^2}fork>0$

The series $\underset{k=1}{\overset{∞}{∑}}{b}_k=\frac{1}{k^2}$is convergent by the p-series test and the series $\underset{k=1}{\overset{∞}{∑}}{a}_k=-\frac{1}{k}$is divergent by the p-series test

If$a_k≤b_k$for every positive integer k and the series $\underset{k=1}{\overset{∞}{∑}}{b}_k$converges , then The series $\underset{k=1}{\overset{∞}{∑}}{a}_k$converge is false

Hence the statement is false.

Consider the statement " if the series $\underset{k=1}{\overset{∞}{∑}}{b}_{k}and\underset{k=1}{\overset{∞}{∑}}{a}_k$both diverge then $\underset{k=1}{\overset{∞}{∑}}(a_k·b_k)$diverge

To determine whether the statement is true or false

Consider the series $\underset{k=1}{\overset{∞}{∑}}{b}_k=\frac{1}{k}and\underset{k=1}{\overset{∞}{∑}}{a}_k=\frac{1}{k}$
$\underset{k=1}{\overset{∞}{∑}}{b}_k=\frac{1}{k}and\underset{k=1}{\overset{∞}{∑}}{a}_k=\frac{1}{k}$are divergent by p-series test.

$\underset{k=1}{\overset{∞}{∑}}(a_k.b_k)=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k^2}$is convergent by the p-series test

Then series $\underset{k=1}{\overset{∞}{∑a_k}}and\underset{k=1}{\overset{∞}{∑}}{b}_k$are convergent by the p-series and $\underset{k=1}{\overset{∞}{∑}}(a_k.b_k)$is not convergent.

Hence the statement is false

Consider the statement " if $a_{k}and{b}_k$are both positive for every positive integer k and $\underset{k→∞}{\lim }\frac{a_k}{b_k}=\frac{1}{2}$then $\underset{k=1}{\overset{∞}{∑}}{b}_{k}and\underset{k=1}{\overset{∞}{∑a_k}}$both converges .

To determine whether given statement is true or false

$a_k=\frac{1}{k}and{b}_k=\frac{2}{k}$

The value of

$\underset{k→∞}{\lim }\frac{a_k}{b_k}=\underset{k→∞}{\lim }\frac{1}{2}=\frac{1}{2}$

$\underset{k=1}{\overset{∞}{∑a_k}}=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k}and\underset{k=1}{\overset{∞}{∑b_k}}=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k}$are both divergent

Hence the statement is false.

Consider the statement : " if $\underset{k=1}{\overset{∞}{∑}}{b}_{k}and\underset{k=1}{\overset{∞}{∑}}{a}_k$both converge, then$\underset{k→∞}{\lim }\frac{a_k}{b_k}$ is finite

To determine whether the statement is true or false

Consider $a_k=\frac{1}{k^2}and{b}_k=\frac{1}{k^3}$

$\underset{k=1}{\overset{∞}{∑}}{a}_k=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k^2}and\underset{k=1}{\overset{∞}{∑}}{b}_k=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k^3}$are both convergent the power series .

The value of $\underset{k→∞}{\lim }\frac{a_k}{b_k}=\underset{k→∞}{\lim }\frac{k^3}{k^2}=\underset{k→∞}{\lim }k=∞$

$\underset{k=1}{\overset{∞}{∑}}{a}_k=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k^2}and\underset{k=1}{\overset{∞}{∑}}{b}_k=\underset{k=1}{\overset{∞}{∑}}\frac{1}{k^3}$are both convergent but it does not have finite limits

Hence the statement is false .