   Chapter 11, Problem 39RE

Chapter
Section
Textbook Problem

# Prove that if the series ∑ n = 1 ∞ a n is absolutely convergent, then the series ∑ n = 1 ∞ ( n + 1 n ) a n is also absolutely convergent.

To determine

To prove: The series n=1(n+1n)an is absolutely convergent.

Explanation

Result used: The Limit Comparison Test

“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.”

Proof:

Consider the given series n=1bn=n=1(n+1n)an

(n+1)an<nan(n+1n)an<an|(n+1n)an|<|an|

Consider the series n=1|an| is greater than n=1|bn|=n=1|(n+1n)an| .

Obtain the limit of limn|anbn|

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