# Need some series/ summation help

1. Aug 11, 2009

### 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.

2. Aug 13, 2009

### soandos

it does not converge...

3. Aug 13, 2009

### g_edgar

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

4. Aug 13, 2009

### 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.

5. Aug 14, 2009

### 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:
$$\sum_{n=1}^{\infty}\;(-1)^{n}\left(\frac{e^{-\frac{1}{nx}}}{n}\right)$$

6. Aug 16, 2009

### rman144

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