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Since $e=\sum_{n=0}^\infty\frac{1}{n!}=1+1+\frac12(1+\frac13(1+\frac14(1+\dots)))$, we have

$$4^{e-2}=\sqrt{4\cdot\sqrt[3]{4\cdot\sqrt[4]{4\cdots}}}$$

Is there however a nice way to express the radical in the title too?

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  • $\begingroup$ I would try to evaluate it to high precision and run the resulting approximation through a reverse symbolic calculator to see if anything comes up. $\endgroup$ Commented Feb 3, 2014 at 22:03
  • $\begingroup$ @BrunoJoyal, I agree it's a good idea, but with one caveat: I ran $4^{e-2}\approx2.70675377667$ through an ISC, just as a "sanity check," and came up empty. $\endgroup$ Commented Feb 3, 2014 at 22:13
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    $\begingroup$ I have to agree, you probably won't find an answer to this. Noting that the power of the radicals changes (or if the values inside change) makes me want to say there is no closed form. $\endgroup$ Commented Feb 1, 2016 at 1:43
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    $\begingroup$ It might be interesting to you that by taking $4$ out of the first root, we obtain another form of this radical: $$2\sqrt{1+\sqrt[3]{\frac{1}{2^4}+\sqrt[4]{\frac{1}{2^{22}}+\dots\sqrt[k]{\frac{1}{2^{k!-2}}+\dots}}}} \approx 2.401615526...$$ You just have to carefully take $1/4$ under each root, moving further and further $\endgroup$
    – Yuriy S
    Commented Jan 16, 2017 at 23:10
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