Find the value of the infinite product ∏k=0∞(1+z2k)\prod_{k=0}^{\infty} (1 + z^{2^k})∏k=0∞(1+z2k) for ∣z∣<1|z| < 1∣z∣<1.
11−z\frac{1}{1-z}1−z1
1−z1-z1−z
11+z\frac{1}{1+z}1+z1
1+z1+z1+z