I A summation series

1. Dec 17, 2017

Gear300

I am asking on the spur, so there has not been too much thought put into it, but how would we classify a series summation such as $$\sum_{i=0}^{n} 2^{2^i} ~ ?$$ It does not feel to be geometric, nor that it can be made to be geometric. In general, the function xx does not look like it bears a Taylor expansion, so I don't think it even has an algebraic approximation (in which case, I doubt it could be the root of a power series either).

I was considering the case |q| < 1, where if we took an infinite product of a summation $$\sum_{i=0}^{n} q^{2^i} ~ ?$$ in some particular way, we might have a convergent power series (or vice versa). Is there already literature on this sort of thing?

Last edited: Dec 17, 2017
2. Dec 17, 2017

Staff: Mentor

3. Dec 17, 2017

Buzz Bloom

This might be relevant:

4. Dec 17, 2017

StoneTemplePython

Your question seems to, curiously, be related to the problem of the week.

specifically, if you take the log of the infinite product to convert to infinite series, recognize you're in radius of convergence for natural log, expand the series (giving you $\sum\sum$), and upper bound $\frac{1}{ 2^{2^n}+2^{-2^n}} \leq \frac{1}{ 2^{2^n}}$ and then interchange the summations...