Virgil
Posts:
8,833
Registered:
1/6/11


Re: Integral test
Posted:
Dec 10, 2012 4:34 PM


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 counterexample. 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 n1/2 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. 

