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I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods suggest it very slowly oscillates toward $0.67$ between $0.65$ and $0.69$. The power tower begins at $\frac{1}{2}$ and is raised to the $\frac{1}{n}$-th power after each iteration: $$\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\frac{1}{5}^\ldots}}}$$ I suspect that there is some way to transform this into some kind of infinite product, or another method that would resolve this, or refining the equation $\frac{W(\ln2)}{\ln2}$. Where $W(x)$ is the Lambert-W function. The given equation would compute the infinite power tower of $\frac{1}{2}$, so one would just find the difference of each successive power and weight it against the initial value of the sum, $\frac{W(\ln2)}{\ln2}$, such that it converges to the desired value by summation. Is there heuristic way to approach this problem?

Tim
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Kyler Rusin
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  • Where does $W(\ln(2))/\ln(2)$ come from? – Gribouillis Jul 24 '23 at 10:02
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    The value seems to oscillate between about $0.658$ and about $0.690$ – Peter Jul 24 '23 at 10:21
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    @Peter Perhaps more precisely between about $0.690347126$ and $0.658365599$ – Henry Jul 24 '23 at 10:49
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    @Gribouillis I suspect the OP is suggesting starting with the tower $(1/2)^{(1/2)^{(1/2)^\cdots}} = W(\ln(2))/\ln(2)$ and then adjusting. But since empirically $(1/2)^{(1/3)^{(1/4)^\cdots}}$ does not converge, that may not work – Henry Jul 24 '23 at 10:57
  • @Gribouillis W(ln(2))/ln(2), comes from the formula -W(-ln(x))/ln(x), which is used the find the converging value of an infinitive power tower, where x = 1/2, so W(ln(2))/ln(2) = (1/2)^(1/2)^(1/2)^… to infinity. – Kyler Rusin Jul 24 '23 at 11:08
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    Based on my numerical calculations, it oscillates between $0.6583655992663312$ (for even $n$) and $0.6903471261149643$ (for odd $n$). – Dan Jul 24 '23 at 16:58
  • Because of other comments here, I am perhaps wrong but for me it is clear at first glance that the limit is $1$. (since all these fraccional powers of $2$ are never less than $1$ and converges to $1$. – Piquito Jul 24 '23 at 18:54
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    I believe the subsequence of odd-height towers converges and the subsequence of even-height towers converges, but the sequence itself doesn't converge. This is probably contained in results proved in Infinite exponentials by Barrow (1936), but it more clearly follows from Theorem 2 on p. 343 in On the convergence of iterated exponentiation---I by Creutz/Sternheimer (1980). See also the references in this MSE answer. – Dave L. Renfro Jul 24 '23 at 19:29
  • @DaveL.Renfro Thanks. – Kyler Rusin Jul 25 '23 at 07:24

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