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.
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A quick check with Mathematica shows that odd terms tend to 0.690347 while even terms tend to 0.658366. – Intelligenti pauca Sep 30 '15 at 19:56
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do you mean the sequence can be nonconvergent ? – Samir Karim Sep 30 '15 at 20:50
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So it seems: I've gone as far as n=1000 and the two subsequences converge quite fast, but to different limits. – Intelligenti pauca Sep 30 '15 at 21:06
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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}]
Intelligenti pauca
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