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

Note by Pi Han Goh
6 years ago

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Comments

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

Dylan Pentland - 6 years ago

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I'm pretty sure it falls between 3.113.11 and 3.133.13.

Pi Han Goh - 6 years ago

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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 Lin(1n){Li}_{-n}(\frac{1}{n}). (The polylogarithm function)

Dylan Pentland - 6 years ago

Wow, this seems to be a pretty interesting problem.....did you make any progress on this @Pi Han Goh sir??

Aaghaz Mahajan - 1 year ago

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No, I don't.

Pi Han Goh - 1 year ago
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