something polylogarithmic for you.

\sum_{n=1}^{\infty} \frac{e^{1/n}}{e^{n}}

\sum_{n=1}^{\infty} \left(\frac{1}{e}\right)^{n} \sum_{m=0}^{\infty} \frac{1}{m!}\frac{1}{n^{m}}

\sum_{m=0}^{\infty} \frac{1}{m!} \sum_{n=1}^{\infty} \left(\frac{1}{e}\right)^{n} \frac{1}{n^{m}}

the inside is just a polylogarithm:

\sum_{m=0}^{\infty} \frac{Li_{m}(1/e)}{m!}

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