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Taking the powerset of the empty set N times, we end up with a set of size 2^2^...^2 (N times). A generating function would then take the form $$1+2x+4x^2+16x^3+65536x^4+ ...$$

I was trying to find a sufficiently differentiable function with such a Taylor series, but then I suspected that such rapidly growing Taylor series cannot possibly exist.

Can one please come up with such a function, or conversely, show that it doesn't exist?

Troy McClure
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    Note that$$2^{2^n}=o\left(\overbrace{2^{2^{\cdots^{2}}}}^{n\text{ twos}}\right)$$but $\sum 2^{2^n}x^n$ has a radius of convergence of $0$ so this series would also have a radius of convergence of $0$. – Peter Foreman Jun 01 '20 at 10:23
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    thanks, right, i'm aware that the radius of convergence must be zero, but we still might be able to study it via asymptotic methods (or analytic continuation, or many other common tricks) – Troy McClure Jun 01 '20 at 10:26

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