Date: Feb 11, 2013 2:24 AM
Author: quasi
Subject: Re: Is this series uniformly convergent for x != 0 ?

quasi wrote:>quasi wrote:>>vv 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.My last claim now seems to be too strong.I think the series diverges for all x except for x = (k*Pi)/d where k,d are integers with k odd and k,d relatively prime.Thus, the series converges pointwise for only countably manyreal values of x. Moreover, even on its countable domain, itseem clear that the convergence is not uniform.quasi