From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: establishing divergence(?) of this series
Date: 11 Nov 1999 17:06:30 -0500
Newsgroups: sci.math
Keywords: Raabe's test

In article <80f6qr$2q2h$1@junkie.gnofn.org>,
Craig Johnston <caj@lfn.org> wrote:
:Take the sum as k ranges from 0 to infinity of (k/e)^k / k!.
:
:Now, Stirling's formula seems to tell me that the general term goes to
:zero and that the series diverges.  (neither one of these things being
:obvious offhand to me)
:
:Anyone know of a way to show this without using Stirling?  I've
:tried all the obvious tests.

There is Raabe's Test for detecting series whose n-th term tends to 0
like n^(-p) (then we know: p>1 makes the series converge, p<1 diverge).
This p can be found as lim(n to infinity) n*(a_n/a_(n+1) - 1) , if the
limit exists.

To get to the limit, you calculate (simplifications are easy)
 ln(a_n / a_(n+1)) = 1 - n * ln(1 + 1/n) = (Maclaurin expansion).

Good luck, ZVK(Slavek).