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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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David Bernier

Posts: 3,892
Registered: 12/13/04
Re: Is this series uniformly convergent for x != 0 ?
Posted: Feb 11, 2013 1:46 AM
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On 02/10/2013 11:35 PM, 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
> Thanks!
> --vv

If x is real, exp(-ixn) = cos(-xn) + i*sin(-xn) .

cos^2(A) + sin^2(A) = 1 if A is real.

If x=0, the series diverges. Also, x = 2kPi , k in Z: divergent.

For x real, it doesn't converge absolutely.

On uniform convergence:

Suppose X is the set of reals where x in X iff
the series converges.

Do we know what X is?


dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.

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