   Chapter 11.4, Problem 26E

Chapter
Section
Textbook Problem

# Determine whether the series converges or diverges.26. ∑ n = 2 ∞ 1 n n 2 − 1

To determine

Whether the series n=21nn21 converges or diverges.

Explanation

Given:

The series is n=21nn21 .

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=21nn21 .

n>n21           [n=n2]1n21  >1n

Divide the above inequality by n on both the sides,

1nn21  >1n2n=21nn21  >n=21n2

Consider the series n=1bn=n=11n2 , which must be smaller than n=1an=n=21nn21

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