   Chapter 11.3, Problem 6E

Chapter
Section
Textbook Problem

Use the Integral Test to determine whether the series is convergent or divergent. ∑ n = 1 ∞ 1 ( 3 n − 1 ) 4

To determine

Whether the series is convergent or divergent

Explanation

1) Concept:

i) Integral test:

Suppose f is a continuous, positive, decreasing function on [1, ) and let an=fn. Then, the 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

ii) Improper integral of infinite intervals:

If atfxdx exists for every number ta, then

af(x) dx=limtatfxdx

provided this limit exists (as a finite number)

2) Given:

n=113n-14

3) Calculation:

According to the concept

an=fn=13n-14

fx=13x-14

For intervals [1, ), the function is positive and continuous

To determine the given function for decreasing, differentiate fx with respect to x,

f'x=-43x-15<0

Therefore, the function is decreasing so the integral test applies

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