Date: Feb 11, 2013 1:10 AM
Author: quasi
Subject: Re: Is this series uniformly convergent for x != 0 ?

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.

quasi