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.
2026-03-29 21:34:21.1774820061
Irrational numbers and series
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