   Chapter 11, Problem 12RE

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.12. ∑ n = 1 ∞ n 2 + 1 n 3 + 1

To determine

Whether the series n=1n2+1n3+1 is convergent or divergent.

Explanation

Given:

The series is n=1n2+1n3+1 .

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+1n3+1

n3+n>n3+11n3+1>1n3+n1n3+1>1n(n2+1)n2+1n3+1>1n

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

Obtain the limit of anbn

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