Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 '14 at 3:56
Yes. That was it. – Mayank Pandey Mar 19 '14 at 4:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.