Could someone please let me know if my steps are correct? I am trying to rewrite the first line as a rational function, $p(x)\over q(x)$
\begin{align} & 3x^1+3x^2+9x^3+9x^4+27x^5+27x^6+\cdots \\[8pt] = {} & 3x(1+x)+9x^3(1+x)+27x^5(1+x)+\cdots \\[8pt] = {} & 3x(1+x)(1+3x^2+9x^4+\cdots) \\[8pt] = {} & (3x+3x^2) {1\over 1-3x^2} \\[8pt] = {} & 3x+3x^2\over 1-3x^2 \end{align}
Your work seems correct to me.