$a_n = (1+ z ^{2^{n-1}})$, where $n$ is a natural number. Prove that $\prod_{n=1}^{\infty} a_n = \frac{1}{1-z}$, where $z\in \mathbf{C} \text{ and } |z| < 1$
Please help me solving this question. I tried to expand the product and use some basic identities to simplify the expansion but none of them worked. Thank you.
$$ (1+x)(1+x^2) = 1+x + x^2 + x^3 $$ that is consecutive exponents from $0$ to $3=4-1.$
$$ (1+x)(1+x^2)(1+x^4) = 1+x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 $$ that is consecutive exponents from $0$ to $7 = 8-1.$
What is $$ (1+x)(1+x^2)(1+x^4)(1+x^8) \; \; \; ? \; \; \; $$