   Chapter 11.5, Problem 12E

Chapter
Section
Textbook Problem

# Test the series for convergence or divergence.12. ∑ n = 1 ∞ ( − 1 ) n + 1 n e − n

To determine

To test: Whether the series is convergent or divergent.

Explanation

Given:

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

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)=xex .

The derivative of the function as follows,

f(x)=ddx(xex)=xddx(ex)+exddx(x)   (by Product Rule)=x(1)ex+ex(1)=ex(1x)

Since f(x)<0 for x>1 and by the Result (2), the given function is a decreasing function

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