# Let $$\displaystyle f(m) =\sum_{n=1}^\infty\frac{n^m}{m^n}$$. Find the real value of $$m$$ for which its corresponding $$f(m)$$ is minimized.

Note by Pi Han Goh
3 years ago

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I don't have a proof yet, but I suspect that the minimum is at $$m=\pi$$.

- 3 years ago

I'm pretty sure it falls between $$3.11$$ and $$3.13$$.

- 3 years ago

Yes, I checked 3.12 and it appears to lower... oh well, that would have been a really cool minimum :(

I don't know if this is really helpful but this is equivalent to minimizing $${Li}_{-n}(\frac{1}{n})$$. (The polylogarithm function)

- 3 years ago

Comment deleted Jun 09, 2015

I didn't restrict $$m$$ to be an integer. A value of $$3.11$$ yields a lower sum.

- 3 years ago