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I wonder if there is a way to get resummation of this series? By this way , i am trying to get the integral representation of this series, it could be by Gamma function.

$$\sum _{k=0}^{\infty} \left [ (-1)^{k}(2k)!\left(\frac {1} {(i+a)^{2k+1}}-\frac {1} {(-i+a)^{2k+1}}\right) \right ]$$

thank you.

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    I don't know what you mean by re-sum. It seems to me that the terms are not going to zero, so the series does not converge.2012-04-26
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    @Gerry: See [resummation](http://en.wikipedia.org/wiki/Resummation), in particular [Borel resummation](http://en.wikipedia.org/wiki/Borel_resummation).2012-04-26
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    @Gerry: I'm guessing he means to ask if there's a way to make the divergent series he has to "make sense", e.g. having $1+2+3+\cdots$ be "equal" to $-\frac1{12}$, for some peculiar definition of "equal"...2012-04-26
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    @J.M.: What are all the scare quotes and the qualification "peculiar" about? Resummation is a perfectly legitimate mathematical tool.2012-04-26
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    @joriki: I know it's legitimate (I've used it a lot), but most people aren't used to it... it's peculiar in that sense.2012-04-26
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    Maybe if you mean what joriki and J. M. think you mean, you could edit that information into your question.2012-04-26
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    @Gerry Motion seconded.2012-04-26
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    Defacing your questions is quite frowned upon; please don't do this.2013-03-27

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