How to evaluate this infinite series?

[ludwig.van]

Hello,

Could anybody help with this series:

$\sum_{n=0}^\infty e^n/(e^n+1)^{a-1},\,\, a>2. $

I tried (without success) to adapt the Riemann integral theorem and the laplace transform.

For the latest, I will need to find the inverse laplace transform of $e^n/(e^n+1)^(a-1)$, which does not seem so straightforward.

Any ideas? thanks.

 

[Office_Shredder]

Hello,

Could anybody help with this series:

\sum_{n=0}^\infty e^n/(e^n+1)^(a-1),\,\, a>2.

I tried (without success) to adapt the Riemann integral theorem and the laplace transform.

For the latest, I will need to find the inverse laplace transform of e^n/(e^n+1)^(a-1), which does not seem so straightforward.

Any ideas? thanks.

You need to use tex and /tex tags instead of dollar signs (I added them here so that other people can read the formula easily). Is it supposed to be this in the summation:
\frac{e^n}{(e^n+1)^{a-1}}

 

[ludwig.van]

Hi Guys,

Is this series really difficult, has anybody tried to evaluate it?

I've tried some ideas but without success:

1) using the laplace transform to evaluate summation (likewise: http://mathdl.maa.org/images/cms_upload/A_Laplace_Transform18380.pdf)

The problem is that in this case I need to find the inverse laplace transform of \frac{e^s}{(e^s+1)^{a-1}}. Any ideas of how to find this inverse laplace transform?

2) using the Riemann integral theorem (that is: http://www.math.nus.edu.sg/~matngtb/Calculus/Riemannsum/Riemannsum.htm ).

I couldn't adjust the terms in order to apply the theorem. Any ideas?

Thanks.

 

[g_edgar]

Why do you think there is a simple way to write this sum?
For most sums, there isn't.

 

[Gib Z]

For a= 3 I got something hideous involving q-digamma functions (http://mathworld.wolfram.com/q-PolygammaFunction.html). For higher fixed integers a=4, 5,6 ... it should be possible to compute them with a similar calculation with some additional differentiations of the sums, but I'd imagine finding a general formula for arbitrary integers > 2 would be difficult to find. If you could find that though, you could conclude it holds for all complex s with real part greater than 2 (by uniqueness of meromorphic continuation).