   Chapter 11.6, Problem 18E

Chapter
Section
Textbook Problem

Use the Ratio Test to determine whether the series is convergent or divergent. ∑ n = 1 ∞ n ! n n

To determine

Whether the series is convergent or divergent by using the ratio test.

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=1n!nn

4) Calculation:

Consider, n=1an=n=1n!nn

Here an=n!nn is positive and an+1=(n+1)!(n+1)n+1

By the ratio test, consider,

limn

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