Need some series/ summation help

[rman144]

$$\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-\frac{1}{nx}}}{n}$$

Where 0<x<oo.

I'm looking for a closed form/ closed representation for this series [I was thinking something like a polylogarithm or dirichlet eta function combination might work].

Any ideas or suggestions would be much appreciated.

 

[soandos]

it does not converge...

 

[g_edgar]

it does not converge...

Sure it does, it differs from $$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ by an absolutely convergent series.

 

[soandos]

If (-1)^n is being raised to $$\frac{e^{-\frac{1}{nx}}}{n}$$, i do not believe it converges. (it can also be simplified, the n's go away). please clarify what you mean.

 

[g_edgar]

I understand the confulsion. If you read the TeX code included, you can see what was actually written. The term to be summed is $(-1)^n$ times a fraction:
$$\sum_{n=1}^{\infty}\;(-1)^{n}\left(\frac{e^{-\frac{1}{nx}}}{n}\right)$$

 

[rman144]

I understand the confulsion. If you read the TeX code included, you can see what was actually written. The term to be summed is $(-1)^n$ times a fraction:
$$\sum_{n=1}^{\infty}\;(-1)^{n}\left(\frac{e^{-\frac{1}{nx}}}{n}\right)$$

Sorry about the confusion. I should have included the brackets as you demonstrated.