
Re: Integral test
Posted:
Dec 11, 2012 5:48 PM


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 counterexample. Any ideas? > > sum_n (1 + (xn)^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((xn)^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

