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Re: Integral test
Posted:
Dec 11, 2012 5:48 PM
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David C. Ullrich wrote:
> 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.
Missing a negative sign before the k(n) ?
Similar would be a sum of bell curves,
sum_n exp(-(x-n)^2 k(n)) ,
where k(n) is, say, n^4 .
> Details in a few days after finals are done, if you remind me...
>
>> Best regards,
>>
>> Jose Carlos Santos
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