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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 x

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?

^{x}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?

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