Originally Posted by

**TheEmptySet** I have been asked to show that for $\displaystyle |z|<1$ that

$\displaystyle \displaystyle \left[ \prod_{j=0}^{\infty}(1-z^{2j+1})\right]^{-1}=\prod_{j=0}^{\infty}(1+z^{j+1})$

I know since $\displaystyle \sum_{j=0}^{\infty}|z^{j+1}|$ and $\displaystyle \sum_{j=0}^{\infty}|z^{2j+1}|$

Converge uniformly on compact subsets of the unit disk that both of the infinite products converge to a holomorphic function of the open disk.

So here is my idea let

$\displaystyle f= \prod_{j=0}^{\infty}(1-z^{2j+1})$ and

$\displaystyle g=\prod_{j=0}^{\infty}(1+z^{j+1})$

It would be sufficient to show that $\displaystyle fg=1$.