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 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).