Date: Dec 11, 2012 12:44 PM
Author: RGVickson@shaw.ca
Subject: Re: Integral test

On Monday, December 10, 2012 10:45:40 PM UTC-8, José Carlos Santos wrote:
> On 10/12/2012 23:15, Butch Malahide wrote:
>
>
>

> >> 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.
>
> >
>
> > Do you mean from [1,+oo) into (0,+oo)?
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>
>
> Yes! My mistake. :-(
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>
>
> Best regards,
>
>
>
> Jose Carlos Santos


The function f(x) = sum_{n=1..infinity} exp(-n^3*(x-n)^2) has sum_n f(n) divergent but int_{x=0..inf} f(x) dx convergent. It has continuous derivatives of all orders at any x, but I don't know if it is analytic.

RGV