Exponential Mean and Beyond

[Catullus]

This sounds terrifyingly hard. Even showing something like this converges sounds terrifyingly hard, lol. Not sure how much luck you'll have, but keep me posted if you think of anything, lol.

Like how hard exactly? Exp time? Exp-Mean(2,3) ~ 2.419. Exp-Mean(2,4) = ln(16)/Lambert_W0(ln(16)) ~ 2.745. Does anyone know of any series for this, like a series for Exp-Mean(x,2)?