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From: Robert Israel on 9 Jun 2010 21:37
On Wed, 9 Jun 2010 05:14:28 -0700 (PDT), zeraoulia wrote:

> On Jun 9, 12:52 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
> wrote:
>> In article
>> <6011a022-4c5c-4e44-a011-bce855788...(a)b35g2000yqi.googlegroups.com>,
>>
>>  zeraoulia <zelhad...(a)gmail.com> wrote:
>>> Hello to all,
>>> My question is: let the alternating serie … (-1)
>>> **n-1*an*exp(i*É¿*ln(n))=0, where (an) is real and i is the complex
>>> number, ln(n) is the logaritm of n. The sum is taken over all natural
>>> numbers n>=1.  I wonder if the above equation (the serie=0) implies
>>> that all the coefficents an=0 for all n. I appreciate if one can help
>>> me to find an answer with some references.
>>
>> 1. Don't use special characters in a text-based newsgroup.
>> What looked like a beta to you looks like gobbledygook once
>> my machine has had its way.
>>
>> 2. It's not an alternating series. That term makes sense only
>> for real series.
>>
>> 3. The (-1)^(n - 1) seems to be superfluous, as it can be absorbed
>> into the a_n, so we have sum a_n exp( i b log n) = 0.
>>
>> 4. What is beta? Is the equation supposed to hold for all beta?
>> or just for one fixed value of beta? is beta real?
>>
>> --
>> Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)
>
> The coefficient (-1)^(n - 1) is not superfluous in the equation. The
> number beta is real and the equation supposed to hold for one fixed
> value of beta.
> Thank you

Then the answer is no. Try a_1 = a_2 = 1, all other a_n = 0,
beta = 2 pi/ln(2).
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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