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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 12, 2012 11:49 AM
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On Tue, 11 Dec 2012 22:48:14 +0000, George Cornelius
<gcornelius@charter.net> wrote:

>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) ?


Yes, sorry.

>
>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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