Showing a statement is true for all positive integers regarding a complex function I am not looking for an answer, for both methods I have one. I'd simply like to check if the statement needs to be proved by induction or simply through rearrangement.
Statement:
For $f(z)=\sum_{n=0}^\infty z^{2^n}$ show that for all positive integers $k$, $f(z)$ satisfies $f(z)=z+z^2+z^4+...+z^{2^{k-1}}+f(z^{2^{k}})$.
My approach:
\begin{align}
z+z^2+z^4+...+z^{2^{k-1}}+f(z^{2^{k}})&=\sum_{n=0}^{k-1} z^{2^n}+\sum_{n=0}^\infty (z^{2^k})^{2^n}\\&=\sum_{n=0}^{k-1} z^{2^n}+\sum_{n=0}^\infty z^{2^{k+n}}\\&=\sum_{n=0}^{k-1} z^{2^n}+\sum_{n=k}^\infty z^{2^n}\\&=\sum_{n=0}^\infty z^{2^n}\\&=f(z)
\end{align}
Would this approach be correct? Or should I prove this statement by induction?

Personally I would have proved it by induction, since "prove this for all positive integers" or whatever often screams that, but your method is valid as well. (And probably the easier of the two, too.)
