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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 0f(x)g(x) for every x >0 and the improper integral 0g(x)dxconverges, then the improper integral 0f(x)dx converges.

b) True or False : if 0f(x)g(x)for every x > 0 and limxf(x)g(x)=3then the improper integrals both 0g(x)dxand0f(x)dxconverge.

c) True or False : if 0ak<1kfor every positive integer k, then the series k=1akconverges .

d) True or False : if 1k2<b kfor every positive integer k, then the series k=1bkdiverges.

e) True or False : if akbkfor every positive integer k and the series k=1bkconverges, then the series k=1akconverges.

f) True or False : if akk=1andk=1bkboth diverge then k=1(ak·bk)diverges .

g) True or False : if akandbkare both positive for every positive integer k and limkakbk=12, then k=1akandk=1bkboth converge.

h) True or False : if akk=1andk=1bkboth converge, then limkakbkis finite .

Short Answer

Expert verified

a) True

b) False

c) False

d) False

e) False

f) False

g) False

h) False

Step by step solution

01

a) step 1

Consider the statement :"if 0f(x)g(x)for every x> 0 and the improper integral 0g(x)dxconverges then the improper integral 0f(x)dx converges".

To determine whether the statement is true or false .

The improper integral 0g(x)dxis convergent .

0g(x)dx=A

Given that 0f(x)g(x)

00f(x)dx0g(x)dx

0f(x)dxA

The improper integral 0f(x)dxconverges

Hence the statement is true.

02

b) step 1: 

consider the statement :" if 0f(x)g(x)for every x>0 andlimxf(x)g(x)=3 then the improper integral 0g(x)dxand0f(x)dxboth converge

To determine whether the statement is true or false

Consider g(x)=1x2andf(x)=3x2

The value of limxf(x)g(x)=limx3=3

03

b) step2 

but the integrals,0g(x)dx=01x2and0f(x)dx=03x2 diverges

Hence the statement is false

04

c) step 1 

Consider the statement : " if 0ak<1kfor every positive integer k, then the series k=1akconverges"

To determine whether the given statement is true or false

using comparison test ,

It states that k=1akandk=1bkbe two series with positive terms such that 0akbkfor every positive integer k.

If the series bkk=1converges , then the series k=1akalso converges .

05

c) step 2

The series k=1bk=k=11kis divergent by the p-series,

The series k=1akis divergent.

Hence the statement is false .

06

d) step1

Consider the statement "if 1k2<bkfor every positive integer k ,then the series k=1bkdiverges .

To determine whether the given statement is true or false.

using comparison test

It states that k=1akandk=1bkbe two series with positive terms such that 0akbkfor every positive integer k . if the series k=1bkconverges then The series k=1akalso converges.

07

d) step 2 

The test fails to determine the converges and diverges of the series k=1bk.

We cannot be said the behavior of k=1bkif 1k2<bkholds

Hence the statement is false .

08

e) step 1

Consider the statement if akbkfor every positive integer k and the series k=1bkconverges , then the series k=1akconverges

To determine whether the statement is true or false.

Consider the series k=1bk=1k2andk=1ak=-1k

Clearly akbkholds as:

-1k<1k2fork>0

09

e) step 2

The series k=1bk=1k2is convergent by the p-series test and the series k=1ak=-1kis divergent by the p-series test

Ifakbkfor every positive integer k and the series k=1bkconverges , then The series k=1akconverge is false

Hence the statement is false.

10

f) step 1

Consider the statement " if the series k=1bkandk=1akboth diverge then k=1(ak·bk)diverge

To determine whether the statement is true or false

Consider the series k=1bk=1kandk=1ak=1k
k=1bk=1kandk=1ak=1kare divergent by p-series test.

k=1(ak.bk)=k=11k2is convergent by the p-series test

Then series akk=1andk=1bkare convergent by the p-series and k=1(ak.bk)is not convergent.

Hence the statement is false

11

g) step 1

Consider the statement " if akandbkare both positive for every positive integer k and limkakbk=12then k=1bkandakk=1both converges .

To determine whether given statement is true or false

ak=1kandbk=2k

The value of

limkakbk=limk12=12

akk=1=k=11kandbkk=1=k=11kare both divergent

Hence the statement is false.

12

h) step 1

Consider the statement : " if k=1bkandk=1akboth converge, thenlimkakbk is finite

To determine whether the statement is true or false

13

h) step 2 

Consider ak=1k2andbk=1k3

k=1ak=k=11k2andk=1bk=k=11k3are both convergent the power series .

14

h) step 3

The value of limkakbk=limkk3k2=limkk=

15

h) step  4

k=1ak=k=11k2andk=1bk=k=11k3are both convergent but it does not have finite limits

Hence the statement is false .

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