TITLE: convergence of a tower power
QUESTION [11 upvotes]: Prove that the sequence of general term $(\frac 12)^{(\frac 13)^{(\frac 14)^{...\frac 1n}}}$ is convergent.
the three dots are antidiagonal of course :)
My try was to compare it with some easier one like the power tower of $\frac 12$.
I conjectured with wolframalpha that it can be divided to two subsequences which seem to be adjacent but i'm finding it hard to prove it
Can anyone give some hints.

REPLY [6 votes]: Here's the result of computing your sequence with Mathematica. It is apparent that odd and even subsequences separately converge to two different limits.
f[n_] := Fold[Power[#2, #1] &, 1, Table[1/k, {k, n, 2, -1}]]

N[f[500], 20]
0.65836559926633118818

N[f[1000], 20]
0.65836559926633118818

N[f[499], 20]
0.69034712611496431947

N[f[999], 20]
0.69034712611496431947

DiscretePlot[f[n], {n, 100}]