Date: Feb 11, 2013 1:15 AM
Author: quasi
Subject: Re: Is this series uniformly convergent for x != 0 ?
quasi wrote:

>vv <vanamali@netzero.net> wrote:

>>

>>I'd be grateful if someone can throw light on whether or not

>>the following series is uniformly convergent for x not equal

>>to zero:

>>

>>\sum_{n=1}^infty exp(-ixn)/n

>

>I could be wrong, but here's what I think ...

>

>If k is a nonzero integer then for x = 2*k*Pi, the series

>diverges.

>

>More generally, I think the series diverges for x = (2*k*Pi)/d

>where k,d are nonzero integers with d odd and with k,d

>relatively prime. Thus, the series is pointwise divergent on a

>dense subset of R, so the question of uniform convergence is

>silly.

>

>In fact, going out on a limb, it seems to me that the series

>diverges for all real numbers x except for x = 0, x = Pi,

>x = -Pi.

I meant: except for x = Pi, x = -Pi.

quasi