   Chapter 11.3, Problem 26E

Chapter
Section
Textbook Problem

Determine whether the series is convergent or divergent. ∑ n − 1 ∞ n n 4 + 1

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 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=1nn4+1

3) Calculation:

According to theconcept

an=fn=nn4+1

fx=xx4+1

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

To determine the given function for decreasing, differentiate f(x) with respect to x

f'x=1-3x4x4+12<0

Therefore, the function is decreasing so the integral test applies.

1xx4+1dx =limt1txx4+1dx=limt1txx22+1dx

Substitute,u=x2

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