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Question Number 67517 by mathmax by abdo last updated on 28/Aug/19

l e t z \in C a n d ∣ z ∣< 1 p r o v e t h a t

\frac{z}{1 - z^{2}} + \frac{z^{2}}{1 - z^{4}} + \dots . . + \frac{z^{2^{n}}}{1 - z^{2^{n + 1}}} + \dots = \frac{z}{1 - z}

\frac{z}{1 + z} + \frac{2 z^{2}}{1 + z^{2}} + \dots . + \frac{2^{n} z^{2^{n}}}{1 + z^{2^{n}}} + \dots . = \frac{z}{1 - z}