   Chapter 11, Problem 21RE

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.21. ∑ n = 1 ∞ ( − 1 ) n − 1 n n + 1

To determine

Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1(1)nnn+1 .

Result used:

(1) “If the alternating series n=1(1)n1bn=b1b2+b3b4+...   bn>0 satisfies the conditions bn+1bn   for all n and limnbn=0 , then the series is convergent; otherwise, the series is divergent.”

(2) The function f(x) decreases when f(x)<0 .

Calculation:

Consider the function from given series, f(x)=xx+1 .

The derivative of the function as follows,

f(x)=ddx(xx+1)=(x+1)ddx(x)xddx(x+1)(x+1)2=(x+1)12(x)121x(1)(x+1)2   [x=x12]=12x12(x+12x)(x+1)2

That is, f(x)=1x2x(x+1)2

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