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Chapter 7: Q.1 b) (page 630)

consider the statement if 0鈮�f(x)鈮�g(x)for every x>0 andlimx鈫�鈭�f(x)g(x)=3, then the improper integrals 鈭�0鈭�g(x)dxand鈭�0鈭�f(x)dxboth converge. The objective is to determine whether statement is true or false.

Short Answer

Expert verified

False

Step by step solution

01

To determine the statement converge or not

Consider the function g(x)=1x2andf(x)=3x2.

The value of limx鈫�鈭�f(x)g(x)is:

limx鈫�鈭�f(x)g(x)=limx鈫�鈭�3

= 3

Therefore, both the integrals, 鈭�0鈭�g(x)dx=鈭�0鈭�1x2dxand鈭�0鈭�f(x)dx=鈭�0鈭�3x2diverges.

Therefore the given statement is false.

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Most popular questions from this chapter

Let 鈭�k=0鈭�crkand鈭�k=0鈭�bvkbe two convergent geometric series. Prove that 鈭�k=0鈭�crk.bvk converges. If neither c nor b is 0, could the series be 鈭�k=0鈭�crk鈭�k=0鈭�bvk?

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0.237237237...

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Consider the series

Fill in the blanks and select the correct word:

Iflimk鈫�鈭�akbk=鈭�and鈭�k=1鈭�_____divergesthen鈭�k=1鈭�_____(converges/diverges).
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