   Chapter 11.6, Problem 33E

Chapter
Section
Textbook Problem

# Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.33. ∑ n = 1 ∞ ( − 9 ) n n 10 n + 1

To determine

Whether the series is absolutely convergent or conditionally 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.”

Calculation:

The given series n=1an=n=1(9)nn10n+1.

Substitute, n=n+1, then an+1=(9)n+1(n+1)10n+2

Obtain the limit of |an+1an|

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