Convergence of infinite product

Question

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 $$[{|z|<1}]$$. What about $$[{|z|>1}]$$.

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.

Answer 1

You should use the definition of absolute convergence of a product: $\prod(1+y_n)$ converges absolutely if and only if $\sum|y_n|$ converges. You can also use the fact that a necessary condition for a product $\prod(1+y_n)$ to converge is that $y_n\to0$. These two statements should allow you to prove the first statement and show that in the second case, the product diverges.