Suppose we have $y=k+ax+bx^2+cx^3...$ as an infinite series. In order to reverse this I have been taught that we can assume $y-k=z$
Let's suppose $x=Az+Bz^2+Cz^3...$ Where $A,B,C...$ are determined by their usual formulas. By substituting $y-k=z$ back we get $x=A(y-k)+B(y-k)^2+C(y-k)^3...$. Rearranging this we obtain $$x=(-Ak+Bk^2-Ck^3....)+y(A-2Bk+3Ck^2...)+y^2(B-3Ck...)$$ Now since each coefficient is an infinite series in itself and for many examples these series may diverge, how do we determine them?
The terms are not an infinite series.
Suppose that we write $$y=k+\sum_{n=1}^{p>5}a_i x^i$$ and we make a Taylor expansion of it up to $O(x^5)$. Now, for more clarity, let $t=\frac{y-k}{a_1}$ and making the series reversion, we should get $$x=t-\frac{a_2 }{a_1}t^2+\frac{\left(2 a_2^2-a_1 a_3\right) }{a_1^2}t^3+\frac{\left(-5 a_2^3+5 a_1 a_3 a_2-a_1^2 a_4\right) }{a_1^3}t^4+O\left(t^5\right)$$