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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
2 years, 4 months ago

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

Dylan Pentland - 2 years, 4 months ago

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I'm pretty sure it falls between \(3.11 \) and \(3.13\).

Pi Han Goh - 2 years, 4 months 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 \({Li}_{-n}(\frac{1}{n})\). (The polylogarithm function)

Dylan Pentland - 2 years, 4 months ago

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Comment deleted Jun 09, 2015

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

Pi Han Goh - 2 years, 4 months ago

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