   Chapter 11.3, Problem 5E

Chapter
Section
Textbook Problem

Use the Integral Test to determine whether the series is convergent or divergent. ∑ n = 1 ∞ 2 5 n − 1

To determine

Whether theseries 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) 1fx dx  is convergent, then n=1an is convergent

b) 1fx dx  is divergent, then n=1an is divergent

ii) Improper integral of infinite intervals:

If atfx dx exists for every number ta, then

af(x) dx=limtatfxdx

provided this limit exists (as a finite number)

2) Given:

n=125n-1

3) Calculation:

According to theconcept,

an=fn=25n-1

fx=25x-1

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

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

f'x=-105x-12<0

Therefore, the function is decreasing so the integral test applies

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