   Chapter 11.6, Problem 3E

Chapter
Section
Textbook Problem

# Determine whether the series is absolutely convergent or conditionally convergent.3. ∑ u − 0 ∞ ( − 1 ) n 5 n + 1

To determine

Whether the series is absolutely convergent or conditionally convergent.

Explanation

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) “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.”

Definition used:

“A series an is called absolutely convergent if the series of absolute values |an| is convergent.”

“A series an is called conditionally convergent if it is convergent but not absolutely convergent.”

Calculation:

Consider the given series n=1an=n=0(1)n5n+1 where an=15n+1>0.

Since 5n+1 is increasing, it can be concluded that 15n+1 is decreasing.

That is, an is decreasing. (1)

Obtain the limit of an.

limnan=limn15n+1=15()+1=1=0

That is, limnan=0

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