K. S. Van Horn gives a few lines describing the derivation in his PT:TLoS errata. I don't understand why he does step 4 there -- it seems to me to be irrelevant. The two main facts which are needed are step 2-3 and step 5, the sum of a geometric series and the Taylor series expansion around y = S(x). Hopefully that is a good hint.

Nitpicking with his errata, 1/(1-z) = 1 + z + O(z^2) for all z is wrong since the interval of convergence for the RHS is (-1,1). This is not important to the problem since the z here will be z = exp(-q) which is less than 1 since q is positive.