\begin{align*} A \amp = 1 + 3x + 5x^2 + 7x^3 + 9x^4 + \cdots\\ \underline{-\qquad xA} \amp \underline{\,\,= 0 + x + 3x^2 + 5x^3 + 7x^4 + 9x^5 + \cdots}\\ (1-x)A \amp = 1 + 2x + 2x^2 + 2x^3 + 2x^4 + \cdots \end{align*}