   Chapter 11.4, Problem 20E

Chapter
Section
Textbook Problem

# Determine whether the series converges or diverges.20. ∑ n = 1 ∞ n 2 + n + 1 n 4 + n 2

To determine

Whether the series n=1n2+n+1n4+n2 converges or diverges.

Explanation

Given:

The series is n=1n2+n+1n4+n2 .

Result used:

(1) “Suppose that an and bn are the series with positive terms, if limnanbn=c , where c is a finite number and c>0 , then either both series converge or both diverge.”

(2) The p-series n=11n is converges if p>1 and diverges if p1 .

Calculation:

The given series is n=1an=n=1n2+n+1n4+n2 .

n2<n2+nn2+1<n2+n+11<n2+n+1n2+1

Divide the inequality by n2 on both the sides,

1n2<n2+n+1n2(n2+1)1n2<n2+n+1n4+n2

Consider the series n=1bn=n=11n2 , which must be smaller than n=1an=n=1n+1n3+n .

Obtain the limit of anbn

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