Re: series and squares
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Sun, 02 Apr 2006 09:07:03 -0500
On Sun, 02 Apr 2006 06:55:11 EDT, eugene <jane1806@xxxxxxx> wrote:
a).Let (a_n) be an increasing sequence of positive integers and let S_n=a_1+a_2+...+a_n.Show that if
liminf(a_n / n) >2+sqrt(2), then for all n sufficiently large there exists a perfect square between S_n and S_{n+1}.
This follows from the fact that if N is a large positive integer then
there exists a perfect square between N and N + ___.
To prove _that_ fact, suppose that n^2 is the largest square with
n^2 <= N. Then (n+1)^2 > N. But (n+1)^2 - N <= (n+1)^2 - n^2, ... .
************************
David C. Ullrich
.
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