   Chapter 11.3, Problem 13E

Chapter
Section
Textbook Problem

Determine whether the series is convergent or divergent. 1 3 + 1 7 + 1 11 + 1 15 + 1 19 + ...

To determine

Whether the series is convergent or divergent

Explanation

1) Concept:

Integral test:

Suppose f is a continuous, positive, decreasing function on [1, ) and let an=fn. Then series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

a) 1fxdx  is convergent, then n=1an is convergent

b) 1fxdx  is divergent, then n=1an is divergent

2) Given:

13+17+111+115+119

3) Calculation:

The given series can be written in nth form as

14n-1 where n=1, 2, 3,

Rewriting the equation in summation form,

=n=114n-1

According to concept,

an=fn=14n-1

fx=14x-1

1fxdx =114x-1dx

114x-1dx =limt1t14x-1dx

Divide and multiply by 4,

=14limt1t44x-1dx

Substitute,u=4x-1

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