Prove that: $$ \prod_{n=1}^{\infty}(1+x^{2^{n-1}})=\dfrac{1}{1-x}\quad (|x|<1) $$
My answer is like this: $$ \dfrac{1}{1-x}=1+x+x^2+\cdots\\=(1+x)+(x^2+x^3)+\cdots\\=(1+x)(1+x^2+\cdots)\\=(1+x)((1+x^2)+(x^4+x^6)+\cdots)\\=(1+x)(1+x^2)(1+x^4+\cdots)\\=\cdots=LHS $$ Is my solution correct and rigorous, if not, how to solve the problem? Thanks!