   Chapter 11.5, Problem 23E

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?23. ∑ n = 1 ∞ ( − 1 ) n + 1 n 6   ( | error | < 0.00005 )

To determine

To show: The series n=1(1)n+1n6 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(1)n+1n6.

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(1)n+1n6, and bn=1n6>0.

n<n+1n6<(n+1)61(n+1)6<1n6bn+1<bn

Since bn+1bn, bn is decreasing

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