Re: a question about a series diversion
From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 07/01/04
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Date: Thu, 01 Jul 2004 15:22:08 -0500
On Fri, 2 Jul 2004 00:58:49 +0900, "tngkr" <s@m.com> wrote:
>I have a question regarding the series:
>1 - M(1)/M(2) + 1 - M(2)/M(3) + 1 - M(3)/M(4) + ....
>,where { M(i) } is an arbitrary strictly increasing sequence of positive
>integers.
Did you really mean that series, or do you want to talk about the
series
(1 - M(1)/M(2)) + (1 - M(2)/M(3)) + (1 - M(3)/M(4)) + ....
? There's a big difference; the terms in the series you
wrote do not tend to 0 regardless of what the M's are,
so the series cannot possibly converge.
>Question: How should one prove that this series converges to infinity.
If you're thinking about the series with the parentheses added,
it can converge or diverge. If you think about it for a second
you'll see that given any sequence a_n with 0 < a_n < 1 there
exists an increasing sequence M(n) such that M(n)/M(n+1) = a_n
for all n. So 1 - M(n)/M(n+1) can also be anything you
want between 0 and 1, hence the sum can be infinite or
any positive number.
************************
David C. Ullrich
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