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

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Jose Carlos Santos

Posts: 4,868
Registered: 12/4/04
Re: Integral test
Posted: Dec 10, 2012 5:10 PM
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On 10-12-2012 21:34, Virgil 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. 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?

>
> 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.


Yes, but no such function exists.

Best regards,

Jose Carlos Santos




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