let-z-C-and-z-lt-1-prove-that-z-1-z-2-z-2-1-z-4-z-2-n-1-z-2-n-1-z-1-z-z-1-z-2z-2-1-z-2-2-n-z-2-n-1-z-2-n-z-1-z-

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

let z ∈C and ∣z∣<1 prove that (z/(1−z^2 )) +(z^2 /(1−z^4 )) +.....+(z^2^n /(1−z^2^(n+1) ))+...=(z/(1−z)) (z/(1+z)) +((2z^2 )/(1+z^2 )) +....+((2^n z^2^n )/(1+z^2^n )) +....=(z/(1−z))

$${let}\:{z}\:\in{C}\:{and}\:\mid{z}\mid<\mathrm{1}\:\:{prove}\:{that} \\ $$$$\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }\:+\frac{{z}^{\mathrm{2}} }{\mathrm{1}−{z}^{\mathrm{4}} }\:+…..+\frac{{z}^{\mathrm{2}^{{n}} } }{\mathrm{1}−{z}^{\mathrm{2}^{{n}+\mathrm{1}} } }+…=\frac{{z}}{\mathrm{1}−{z}} \\ $$$$\frac{{z}}{\mathrm{1}+{z}}\:+\frac{\mathrm{2}{z}^{\mathrm{2}} }{\mathrm{1}+{z}^{\mathrm{2}} }\:+….+\frac{\mathrm{2}^{{n}} \:{z}^{\mathrm{2}^{{n}} } }{\mathrm{1}+\mathrm{z}^{\mathrm{2}^{\mathrm{n}} } }\:+….=\frac{\mathrm{z}}{\mathrm{1}−\mathrm{z}} \\ $$

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