# Minimize $f(m)=\sum_{n=1}^\infty n^m / m^n$

For what real value of $m$ such that $\displaystyle\sum_{n=1}^\infty \frac{n^m}{m^n}$ is minimized?

I've been told that it's equivalent of solving for $\text{Li}_{-n}\left(\frac1n\right)$ for the polylogarithmic function.

Is there a way to solve this other than by plug and chug?

• I have changed the formatting of the title so as to make it take up less vertical space -- this is a policy to ensure that the scarce space on the main page is distributed evenly over the questions. See here for more information. Please take this into consideration for future questions. Thanks in advance. Jun 13 '15 at 16:50
• Thank you AlexR, I didn't know of such a policy. Jun 13 '15 at 16:51