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Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational when evaluated at 1. How would one prove this? Taking $\ln$ of this results in a sum which is nicer, but does not offer any immediate insights into the problem, and doesn't easily generalize to higher derivatives.

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This function is only defined at $x=1$ and $x=0$ (the latter depending on the definition of infinite product.) So it can't have any derivatives. Perhaps you mean $\prod (1+x2^{-n})$? –  Thomas Andrews Mar 19 at 3:56
    
Yes. That was it. –  Mayank Pandey Mar 19 at 4:16

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