From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: divergence,limits etc...
Date: Mon, 1 Nov 1999 17:22:59 -0600 (CST)
Newsgroups: sci.math
To: hovdan@online.no
Keywords: [missing]

In article <381E1985.B93A089@online.no> you write:
>
>Mike wrote:
>
>> Let's say a(n) is a row.We know that this row is divergent
>> and limited and that lim(a(n+1)/a(n))=1 (when n tends to
>> infinit)
>> Can you find such thing? If you do please tell me!
>
>I can't see how it can be divergent when lim(a(n+1))=lim(a(n))...?

No one said lim(a(n)) existed! 

Try e.g.  a(n) = 2 + sin(H(n))  where  H(n) = 1 + 1/2 + 1/3 + ...
We can show  lim(a(n+1)/a(n)) = 1 by showing the log of the ratio goes
to zero. This is the limit of the differences of the  b(n) = log(a(n)),
but  b(n+1) - b(n)  is roughly  b'(n) = a'(n)/a(n) = cos(H(n))*H'(n)/a(n)
~ (1/n) * ( cos(H(n)) / a(n) )  . The first factor drops to zero while
the second stays bounded. With some effort this can be made precise.

I won't post this until the original person's homework is submitted :-)

dave