Raabe's Theorem

Date: 8/23/96 at 4:49:25
From: Manuel Ojeda Aciego
Subject: about Raabe's theorem

Dear Dr. Math:

This question is about Raabe's result for the convergence of a
numerical series of non-negative terms.

It states the convergence or divergence of the series by means of the
result m of the following limit
                               a(n+1)
             lim     n ( 1 -  --------  ) = m.
            n->Infty            a(n)

If m > 1 then the series converge, if m < 1 then the series diverge, 
and (here is my problem) if m = 1 nothing can be asserted.

The harmonic series is an example of divergent series having m=1,
what I am looking for is a convergent series having m = 1,
in order to convince myself of the last part of the theorem.

Best regards,

Manuel Ojeda Aciego                              
Dept. Matematica Aplicada                      
Universidad de Malaga                              

Date: 8/23/96 at 19:39:50
From: Doctor Tom
Subject: Re: about Raabe's theorem

How about a(n) = 1/(n*log(n)*log(n))?

This converges (See Knopp, Infinite Sequences and Series, for 
example), and with a little work, I think you can convince yourself 
that the limit of n(1 - a(n+1)/a(n)) is also 1.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/