R help - Sep 2005 - (Off topic) Limit question.

Please reply to me directly (rolf at math.unb.ca) rather than to the
list, since the question is completely R-free and I'm simply asking
this list because there are so many clever and knowledgeable people
on it.

Suppose that n_i, i = 1, 2, 3, ... are positive integers, and that

                      k
	  lim        SUM n_i^j = nu_j < infinity
     k --> infinity  i=1

for j = 1, 2, 3.  Need it be the case that

                     k-1
	  lim        SUM n_i * n_{i+1}     exists?
     k --> infinity  i=1

I can neither prove this, nor come up with a counter-example.
Can anyone help me out?

				cheers,

					Rolf Turner