Associated Legendre Equation truncation

Question

In trying to solve the ALE for the Schrödinger Equation through a power series I have run into a difficulty $$(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+(A-\frac{m^2}{1-x^2})y=0$$ has as a solution $$=(1-x^2)^\frac{|m|}{2}\sum_{n=0}^\infty a_n x^n$$ The recursion relation is then $$\frac{a_{n+2}}{a_n}=\frac{(n+|m|)(n+|m|+1)-A}{(n+1)(n+2)}$$ Now it this series doesn't terminate, for high values of $n$ we have approximately $$\frac{a_{n+2}}{a_n}=1$$ So the series becomes as $$\frac{1}{1-x^2}$$ But then y only seems to diverge if $|m|=1$, while it converges for larger values. Whta am I missing?