Q.1 - Calculus

Question

Determine whether each of the statements that follow true or false . if a statement is true, explain why. if a statement is false , provide a counter example .

a) True or False : if  $0\le f\left(x\right)\le g\left(x\right)$  for every x >0 and the improper integral ${\int }_0^{\infty }g\left(x\right)dx$ converges, then the improper integral ${\int }_0^{\infty }f\left(x\right)dx$  converges.

b) True or False : if  $0\le f\left(x\right)\le g\left(x\right)$ for every x > 0 and $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=3$ then the improper integrals  both ${\int }_0^{\infty }g\left(x\right)dx and {\int }_0^{\infty }f\left(x\right)dx$ converge.

c) True or False : if $0\le a_k<\frac{1}{k}$for every positive integer k, then the series $\sum _{k=1}^{\infty }{a}_k$converges .

d) True or False : if $\frac{1}{k^2}$<b _k for every positive integer k, then the series $\sum _{k=1}^{\infty }{b}_k$diverges.

e) True or False : if $a_k\le b_k$ for every positive integer k  and the series $\sum _{k=1}^{\infty }{b}_k$ converges, then the series  $\sum _{k=1}^{\infty }{a}_k$ converges.

f) True or False : if  $\underset{k=1}{\overset{\infty }{\sum a_k}} and \sum _{k=1}^{\infty }{b}_k$ both diverge then  $\sum _{k=1}^{\infty }(a_k·b_k)$ diverges .

g) True or False : if $a_k and b_k$ are both positive for every positive integer k and $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}$=$\frac{1}{2}$ , then $\sum _{k=1}^{\infty }{a}_k and \sum _{k=1}^{\infty }{b}_k$ both converge.

h) True or False : if $\underset{k=1}{\overset{\infty }{\sum a_k}} and \sum _{k=1}^{\infty }{b}_k$ both converge, then $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}$ is finite .

Step-by-Step Solution

Answer

a) True

b) False 

c) False

d) False

e) False 

f) False 

g) False

h) False 

a) step 1

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

To determine whether the statement is true or false .

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

${\int }_0^{\infty }g\left(x\right) dx=A$

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

$0\le {\int }_0^{\infty }f\left(x\right)dx\le {\int }_0^{\infty }g\left(x\right)dx$

${\int }_0^{\infty }f\left(x\right)dx\le A$

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

Hence the statement is true.

b) step 1:

consider the statement :" if $0\le f\left(x\right)\le g\left(x\right)$ for every x>0 and$\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=3$  then the improper integral  ${\int }_0^{\infty }g\left(x\right)dx and {\int }_0^{\infty }f\left(x\right)dx$both converge

To determine whether the statement is true or false 

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

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

b) step2

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

Hence the statement is false 

c) step 1

Consider the statement : " if $0\le a_k<\frac{1}{k}$for every positive integer k, then the series $\sum _{k=1}^{\infty }{a}_k$ converges"

To determine whether the given statement is true or false 

using comparison test ,

It states that $\sum _{k=1}^{\infty }{a}_k and \sum _{k=1}^{\infty }{b}_k$ be two series with positive terms such that $0\le a_k\le b_k$ for every positive integer k.

If the series $\underset{k=1}{\overset{\infty }{\sum b_k}}$converges , then the series $\sum _{k=1}^{\infty }{a}_k$ also converges .

c) step 2

The series $\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k}$ is divergent by the p-series,

The series $\sum _{k=1}^{\infty }{a}_k$ is divergent.

Hence the statement is false .

d) step1

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

To determine whether the given statement is true or false.

using comparison test

It states that $\sum _{k=1}^{\infty }{a}_k and \sum _{k=1}^{\infty }{b}_k$ be two series with positive terms such that $0\le a_k\le b_k$ for every positive integer k . if the series $\sum _{k=1}^{\infty }{b}_k$ converges then The series $\sum _{k=1}^{\infty }{a}_k$ also converges.

d) step 2

The test fails to determine the converges and diverges of the series $\sum _{k=1}^{\infty }{b}_k$.

We cannot be said the behavior of $\sum _{k=1}^{\infty }{b}_k$if $\frac{1}{k^2}<b_k$holds

Hence the statement is false .

e) step 1

Consider the statement if $a_k\le b_k$for every positive integer k and the series $\sum _{k=1}^{\infty }{b}_k$ converges , then the series $\sum _{k=1}^{\infty }{a}_k$ converges

To determine whether the statement is true or false.

Consider the series $\sum _{k=1}^{\infty }{b}_k=\frac{1}{k^2} and \sum _{k=1}^{\infty }{a}_k=-\frac{1}{k}$

Clearly $a_k\le b_k$ holds as:

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

e) step 2

The series $\sum _{k=1}^{\infty }{b}_k=\frac{1}{k^2}$is convergent by the p-series test and the series $\sum _{k=1}^{\infty }{a}_k=-\frac{1}{k}$is divergent by the p-series test

If$a_k\le b_k$ for every positive integer  k and the series $\sum _{k=1}^{\infty }{b}_k$converges , then The series $\sum _{k=1}^{\infty }{a}_k$converge is false

Hence the statement is false.

f) step 1

Consider the statement " if the series $\sum _{k=1}^{\infty }{b}_{k } and \sum _{k=1}^{\infty }{a}_k$ both diverge then $\sum _{k=1}^{\infty }(a_k·b_k)$diverge

To determine whether the statement is true or false 

Consider the series $\sum _{k=1}^{\infty }{b}_k=\frac{1}{k} and \sum _{k=1}^{\infty }{a}_k=\frac{1}{k}$
$\sum _{k=1}^{\infty }{b}_k=\frac{1}{k} and \sum _{k=1}^{\infty }{a}_k= \frac{1}{k}$ are divergent by p-series test.

$\sum _{k=1}^{\infty }(a_k.b_k)=\sum _{k=1}^{\infty }\frac{1}{k^2}$ is convergent by the p-series test

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

Hence the statement is false

g) step 1

Consider the statement " if $a_{k } and b_k$ are both positive for every positive integer k and $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\frac{1}{2}$ then $\sum _{k=1}^{\infty }{b}_k and \underset{k=1}{\overset{\infty }{\sum a_k}}$ 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 

$\underset{k\to \infty }{\lim } \frac{a_k}{b_k} =\underset{k\to \infty }{\lim }\frac{1}{2}=\frac{1}{2}$

$\underset{k=1}{\overset{\infty }{\sum a_k}}=\sum _{k=1}^{\infty }\frac{1}{k} and \underset{k=1}{\overset{\infty }{\sum b_k}} =\sum _{k=1}^{\infty }\frac{1}{k}$ are both divergent

Hence the statement is false.

h) step 1

Consider the statement : " if  $\sum _{k=1}^{\infty }{b}_k and \sum _{k=1}^{\infty }{a}_k$both converge, then$\underset{k\to \infty }{\lim }\frac{a_k}{b_k}$  is finite

To determine whether the statement is true or false 

h) step 2

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

$\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k^2} and \sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k^3}$ are both convergent the power series .

h) step 3

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

h) step 4

$\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k^2} and \sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k^3}$ are  both convergent but it does not have finite limits

Hence the statement is false .