Re: A strange series

From: georgesZ (zellerg_at_wanadoo.fr)
Date: 06/26/04 

Date: 26 Jun 2004 09:37:32 -0700

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).

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

Friendly,
Georges

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