   Chapter 11.5, Problem 26E

Chapter
Section
Textbook Problem

# Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?26. ∑ n = 1 ∞ ( − 1 n ) n   ( | error | < 0.00005 )

To determine

To show: The series n=1(1n)n is convergent; find the number of terms of the series to be added to the sum to the indicated accuracy.

Explanation

Given:

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

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.

(2) “If s=(1)n1bn, where bn>0, is the sum of an alternating series that satisfies the conditions, bn+1bn and limnbn=0, then |Rn|=|ssn||bn+1|.”

Proof:

Consider the given series n=1(1n)n=n=1(1)n1nn, and bn=1nn>0.

(n+1)n+1>(n)n+1(n+1)n+1>(n)n1(n)n>1(n+1)n+1

Since bn+1bn, bn is decreasing

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