Combining binomial series

A quotient can be written as a product with a negative exponent: \(\frac{\sqrt{1+ax}}{1-bx}=(1+ax)^{1/2}(1-bx)^{-1}\).

Expand each factor to the required degree, multiply systematically and collect like powers. If the multiplier contains only non-negative powers, terms of higher degree cannot contribute to a lower-degree coefficient.

If \(P=1+ax+bx^2+\cdots\) and \(Q=1+cx+dx^2+\cdots\), then \(PQ=1+(a+c)x+(b+ac+d)x^2+\cdots\).

After cancellation, distinguish the first three non-zero terms from all terms up to \(x^3\). They need not describe the same truncation.

For a general coefficient in \((1-kx)^{-2}\), use \((1-kx)^{-2}=\sum_{r=0}^{\infty}(r+1)k^r x^r\), valid for \(|kx|<1\).