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I A summation series

  1. Dec 17, 2017 #1
    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
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  3. Dec 17, 2017 #2

    fresh_42

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  4. Dec 17, 2017 #3

    Buzz Bloom

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  5. Dec 17, 2017 #4

    StoneTemplePython

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

    https://www.physicsforums.com/threads/intermediate-math-problem-of-the-week-12-11-2017.934137/

    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...
     
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