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This could be something which is already somewhere in the website, but I am unable to locate any.

Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of $\biggr[{|z|<1}\biggr]$.

What about $\biggr[{|z|>1}\biggr]$?

I have a feeling this need to be done with using some logarithm expansion and using Weierstrass theorem. But I do like to see more ideas of proof. I am not sure on using those ideas either.

  • 0
    Try taking $z=1+\epsilon$. What happens to the size of the terms in the product?2012-12-23
  • 0
    For more references see ['Euler function'](http://en.wikipedia.org/wiki/Euler_function).2012-12-23

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