sqrt thingy at MSE

[bo198214]

Very interesting, thank you!
Recently, I posted another (and maybe easier) question on the same line that I would like to be answered: convergence value of this series.
I suspect that the constant above could be very close to \( e^{\frac{1}{e}} \), but I cannot find the exact value... it would be interesting since it compares (asymptotically) a unitary increment on hyper-2 to a unitary increment on hyper-4.

Its not an exact proof, but we can take for granted that \(f_m(x):=^{m+1}x-^mx\to\infty\) for \(m\to\infty\) if \(x>e^{\frac{1}{e}}=:\eta\) and \(f_m(x)\to 0\) for \(1<x\le\eta\). Also that it is strictly increasing for each m.
Hence there is an M, such that \(f_M(\eta)<1\) and can not be a solution for all \(x\le \eta\) and \(m\ge M\). 
On the other hand for each \(x_1>\eta\) there is another M such that \(f_m(x)>1\) for all \(x\ge x_1\) and \(m\ge M\).
So for each \(x_2>\eta\) there is an m such that the solution of \(f_m(x)=1\) is in the open interval \((\eta,x_2)\).
So the limit is \(\eta\).