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1/(1-z) = lim }(1 + z)(1 + z^2)(1 + z^{2^2} ...

Source: Spanish Mathematical Olympiad 1971 P5

December 5, 2022
complex numberslimitanalysis

Problem Statement

Prove that whatever the complex number zz is, it is true that (1+z2n)(1−z2n)=1−z2n+1.(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}. Writing the equalities that result from giving nn the values 0,1,2,...0, 1, 2, . . . and multiplying them, show that for ∣z∣<1|z| < 1 holds 11−z=lim⁡k→∞(1+z)(1+z2)(1+z22)...(1+z2k).\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).
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