Show that $(1+z)\Pi^{\infty}(1+z^{2^{n}})=\frac{1}{1-z}$ for $z \in \mathbb{C}$ $|z|<1$

Question

$(1+z)\Pi^{\infty}(1+z^{2^{n}})=\frac{1}{1-z}$ for $z \in \mathbb{C}$ $|z|<1$.

Any tricks to prove this? I'm not really sure how to start.

Answer 1

Look at the partial products. You should be able to prove by induction that $$\prod_{k=0}^n(1+z^{2^k})=\sum_{k=0}^{2^{n+1}-1}z^k.$$