Probar que si $|z|<1$ $\prod_{k=0}^\infty (1+z^{2^k})$ converge y es igual a $(1-z)^{-1}$.
Mi intento:
Tenga en cuenta que
\begin{equation}
\begin{aligned}
(1-z)\prod_{k=0}^N (1+z^{2^k}) &= (1-z)(1+z)(1+z^2)(1+z^{2^2})\cdots(1+z^{2^N})\\
&= (1-z^2)(1+z^2)(1+z^{2^2})(1+z^{2^3})\cdots(1+z^{2^N})\\
&= (1-z^{2^2})(1+z^{2^2})(1+z^{2^3})(1+z^{2^4})\cdots(1+z^{2^N})\\
&= (1-z^{2^{N+1}})
\end{aligned}
\end{equation}
Para |z|<1
\begin{equation}
\lim_{N\to\infty} (1-z)\prod_{k=0}^N (1+z^{2^k})= \lim_{N\to\infty}(1-z^{2^{N+1}}) =1.
\end{equation}
Es esto suficiente para resolver el problema?
