   Chapter 11.6, Problem 24E

Chapter
Section
Textbook Problem

# Use the Ratio Test to determine whether the series is convergent or divergent.24. ∑ n = 1 ∞ ( − 1 ) n 2 n n ! 5 ⋅ 8 ⋅ 11 ⋅ ⋯ ⋅ ( 3 n + 2 )

To determine

Whether the series is convergent or divergent.

Explanation

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.”

Theorem:

“If a series an is absolutely convergent, then it is convergent.”

Calculation:

The given series n=1(1)n2nn!5811(3n+2).

Here, the nth term is an=(1)n2nn!5811(3n+2).

Substitute n=n+1 in an,

an+1=(1)n+12n+1(n+1)!5811(3n+2)(3(n+1)+2)an+1=(1)n+12n+1(n+1)!5811(3n+2)(3n+5)

Obtain the limit of |an+1an|

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