Re: A strange series

From: Simeon Stefanov (sim_stef_at_yahoo.com)
Date: 06/27/04 

Date: 26 Jun 2004 22:14:13 -0700

zellerg@wanadoo.fr (georgesZ) wrote in message news:<5227cf44.0406260837.2b67b03f@posting.google.com>...
> sim_stef@yahoo.com (Simeon Stefanov) wrote in message
> > By the way, here is a continuous analogue:
> > Let f(x) be a smooth function in [1, infinity) such that
> > i) f(x) > 0,
> > ii) f(x)-->infty, as x-->infty
> > iii) |f'(x)|<M.
> > Then the integral
> > Integral [x from 1 to infty] [f(x) / F(x)]^2 dx
> > is convergent, where
> > F(x)= Integral [t from 1 to x] f(t) dt.
>
> There is a problem for 1 (take f(x)=x).

OK, the integral should be from 2 to infinity.

> Could it be :
> Let f(x) be a smooth function in [1, infinity) such that
> i) f(1) > 0,

I suppose, you mean f(x) > 0, otherwise F(x) may vanish and the
integral to diverge.

> iii) |f'(x)|<M.
> Then the integral
> Integral [x from 2 to infty] [f(x) / F(x)]^2 dx
> is convergent, where
> F(x)= Integral [t from 1 to x] f(t) dt?

It is possible that condition ii) f(x)-->infty is unnecessary, I added
it by analogy with the original question.
>
> Friendly,
> Georges

Simeon

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