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

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Virgil

Posts: 9,012
Registered: 1/6/11
Re: Integral test
Posted: Dec 10, 2012 4:34 PM
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In article <ain14pFeqhoU1@mid.individual.net>,
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? I don't think so, but I was unable
> to find a counter-example. Any ideas?
>
> Best regards,
>
> Jose Carlos Santos


One can imagine an analytic function which is equal to 1 at every
natural number but such that the sequence of its integrals from n-1/2 to
n+1/2 converges.

I do not have a concrete example in mind but I'm certain that it is
possible.

It could easily be derived from an analytic function with value 0
outside [-0.5 , .5] and value 1 at 0.

Then the integral from 1 to oo would converge but the sequence f(n)
would not.
--





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