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Topic: Is this series uniformly convergent for x != 0 ?
Replies: 5   Last Post: Feb 11, 2013 5:28 AM

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quasi

Posts: 10,307
Registered: 7/15/05
Re: Is this series uniformly convergent for x != 0 ?
Posted: Feb 11, 2013 1:15 AM
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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



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