If some terms are negative, take absolute values. The Absolute Convergence Test allows this: if the new series converges, then the old one will also.

The Ratio Test will usually cause a lot of cancellation in these cases; cancellation which will rid you of most, if not all, of the factorial part.

For instance, look at:

the sum over
n from 1 to infinity of (n!)/(n^n)

The Ratio Test gives


(n+1)! n^n

(n+1) n^n
(n+1) (n+1)^n

n^n/(n+1)^n = [n/(n+1)]^n --> 1/e as n->infinity

The reason is that [(n+1)/n)]^n=[1+1/n]^n -> e as n->infinity. This is by definition of the exponential.

Hence, according to the Ratio Test the original series is convergent.

Copyright © 1996 Department of Mathematics, Oregon State University

If you have questions or comments, don't hestitate to contact us.