quasi
Posts:
10,662
Registered:
7/15/05


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


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

