[R] (Off topic) Limit question --- corrected!!!

[Rolf Turner]

It was gently pointed out to me by Ted Harding that my question was a
load of dingos' kidneys.  What happened was that I left out a crucial
factor of 1/k.

Here's the question again, stated correctly this time.
(I think!!!)

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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

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

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

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

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

				cheers,

					Rolf Turner