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Topic: Integral test
Replies: 17   Last Post: Dec 20, 2012 12:29 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Integral test
Posted: Dec 11, 2012 1:37 PM
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On Mon, 10 Dec 2012 21:05:31 +0000, José Carlos Santos
<jcsantos@fc.up.pt> wrote:

>Hi all,
>One of my students asked me today a question that I was unable to
>answer. Let _f_ be an analytical function from (0,+oo) into [1,+oo) and
>suppose that the integral of _f_ from 1 to +oo converges. Does it follow
>that the series sum_n f(n) converges?

Certainly not.

> I don't think so, but I was unable
>to find a counter-example. Any ideas?

sum_n (1 + (x-n)^2)^{k(n)}

gives a counterexample if k(n) -> infinity
fast enough.

Details in a few days after finals are done, if you remind me...

>Best regards,
>Jose Carlos Santos

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