# Convergence of infinite product

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.

-
Try taking $z=1+\epsilon$. What happens to the size of the terms in the product? –  Eric Naslund Dec 23 '12 at 5:03
For more references see 'Euler function'. –  Raymond Manzoni Dec 23 '12 at 9:10

## 1 Answer

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.

-
So the hint given by @Eric Naslund make sense now. Thanks Greg Martin. –  Deepak Dec 23 '12 at 6:59
You're welcome! This is a good example to show why the fundamentals of a subject are so important. We can't treat every problem like a brand new mystery unto itself; we need to build up basic techniques to run new problems by, to sort the easily solved ones from the ones that require deeper though. –  Greg Martin Dec 23 '12 at 14:50