   Chapter 11.4, Problem 19E

Chapter
Section
Textbook Problem

# Determine whether the series converges or diverges.19. ∑ n = 1 ∞ n + 1 n 3 + n

To determine

Whether the series n=1n+1n3+n converges or diverges.

Explanation

Given:

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

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:

Consider n=1an=n=1n+1n3+n .

Then, n<n2 .

n+n3<n2+n3n+n3<n2(n+1)1n2<(n+1)n+n3

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

Obtain the limit of anbn

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