   Chapter 11.4, Problem 27E

Chapter
Section
Textbook Problem

# Determine whether the series converges or diverges.27. ∑ n = 1 ∞ ( 1 + 1 n ) 2 e − n

To determine

Whether the series n=1(1+1n)2en converges or diverges.

Explanation

Given:

The series is n=1(1+1n)2en .

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 geometric series n=1arn1 is convergent if |r|<1 and divergent if |r|>1 .

Calculation:

The given series is n=1an=n=1(1+1n)2en .

(1+1n)2en>en         [(1+1n)2>0]n=1(1+1n)2en>n=1en

Consider the series n=1bn=n=1en , which must be smaller than n=1an=n=1(1+1n)2en

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