Let \[x=1+3a+6a^2+10a^3+\dots\] and \[y=1+4b+10b^2+20b^3+\dots\] for \(\lvert a\rvert<1\) and \(\lvert b\rvert<1\). Find \[S=1+3(ab)+5(ab)^2+\dots\] in terms of \(x\) and \(y\).
I found \(x\) and \(y\) in terms of \(a\) and \(b\) respectively, and then substituted in a closed expression for \(S\), which came out to be pretty complicated. This leads me to think that I'm wrong somewhere. Also, is there a method faster than finding closed expressions for \(x\) and \(y\) and then substituting or whatever?