   Chapter 11.6, Problem 42E

Chapter
Section
Textbook Problem

# Let {bn} be a sequence of positive numbers that converges to 1 2 . Determine whether the given series is absolutely convergent.42. ∑ n − 1 ∞ ( − 1 ) n n ! n n b 1 b 2 b 3 ⋯ b n

To determine

Whether the series absolutely convergent or not.

Explanation

Given:

A sequence of positive numbers {bn} converges to 12.

Result used: The Ratio Test

“(i) If limn|an+1an|=L<1, then the series n=1an is absolutely convergent (and therefore convergent.)

(ii) If limn|an+1an|=L>1 or limn|an+1an|=, then the series n=1an is divergent.

(ii) If limn|an+1an|=1, the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an.”

Calculation:

The given series n=1an=n=1(1)nn!nnb1b2b3bn.

Substitute, n=n+1, then an+1=(1)n+1(n+1)!(n+1)n+1b1b2b3bnbn+1

Obtain the limit of |an+1an|.

limn|an+1an|=limn|(1)n+1(n+1)!(n+1)n+1b1b2b3bnbn+1(<

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