To show two formal power series equal

Question

I am wondering whether the following two formal power series are equal: $A(x)=\Pi_{k=1}^{\infty}\frac{1}{1-x^{2k-1}}$, $B(x)=\Pi_{k=1}^{\infty}(1+x^k)$.

Answer 1

We obtain \begin{align*} \color{blue}{\prod_{k=1}^\infty(1+x^k)}&=\prod_{k=1}^\infty\frac{(1+x^k)(1-x^k)}{1-x^k}=\prod_{k=1}^\infty\frac{1-x^{2k}}{1-x^k}\color{blue}{=\prod_{k=1}^\infty\frac{1}{1-x^{2k-1}}} \end{align*}