   Chapter 11.6, Problem 33E

Chapter
Section
Textbook Problem

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

To determine

Whether the given series is absolutely convergent, conditionally convergent or divergent.

Explanation

1) Concept:

Use the ratio test.

2) The Ratio Test:

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

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

(iii) If limnan+1an=1, then the ratio test is inconclusive, that is, no conclusion can be drawn about the convergence or divergence of an.

3) Given:

n=1-9nn 10n+1

4) Calculation:

Consider,n=1an=n=1-9nn 10n+1

Here, an=-9nn 10n+1 is positive and an+1=-9n+1(n+1) 10n+2

By the ratio t

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