# Divergence of an infinite product

How can I prove that the infinite product

$$\displaystyle\prod_{n=1}^{+\infty}(1+z^{2n})$$

diverges for $|z|>1$?

$|1+z^{2n}|\geqslant |z|^{2n}-1\to +\infty$, so the partial products cannot converge.
I thought I should mention the nice identity $$\prod_{k=0}^\infty(1+x^{2^k})=\frac{1}{1-x}\text{ , for all x < 1}$$