Waste less time on Facebook — follow Brilliant.
×

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
1 year, 11 months ago

No vote yet
1 vote

Comments

Sort by:

Top Newest

I don't have a proof yet, but I suspect that the minimum is at \(m=\pi\). Dylan Pentland · 1 year, 11 months ago

Log in to reply

@Dylan Pentland I'm pretty sure it falls between \(3.11 \) and \(3.13\). Pi Han Goh · 1 year, 11 months ago

Log in to reply

@Pi Han Goh 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 · 1 year, 11 months ago

Log in to reply

Comment deleted Jun 09, 2015

Log in to reply

@Joel Yip I didn't restrict \(m\) to be an integer. A value of \(3.11\) yields a lower sum. Pi Han Goh · 1 year, 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...