[MO] Residue at ∞ and ∑(-1)^n x^(2^(2^n))

[Caleb]

Not to sound like a broken record, but:

*touches nose*

I appears most of this works out as I predicted, the Cauchy's integral theorem for fractional derivatives does give the desired result. I recently stubled upon this paper which proves that a neccesary AND sufficent condition for analytical continuation across arcs for a power series \( \sum f_n x^n \) is that \( f_n \) can be extended to an entire function of exponential type and that a_n is given by that cauchy integral theorem I mentioned before. I found this result from this MO answer: https://mathoverflow.net/a/369847/146528