Consider the initial value problem, $$u''+(x+1)u'-u=0 \ \ \ \ \ \ u(-1)=2, u'(-1)=0.$$ I wish to find the radius of convergence of the power series solution (about $x=-1$).
My attempt:
I have found that the recurrence relation is $$A_{k+2}=-\frac{(k-1)A_k}{(k+2)(k+1)} \ \ k\geq 0$$ To find the radius of convergence, I use the ratio test (where $z=x+1$), $$\lim_{k\to\infty}\left|\frac{A_{k+2}z^{k+2}}{A_kz^k}\right|=\lim_{k\to\infty}\left|\frac{1-k}{(k+2)(k+1)}\right||z^2|=0.$$ Now by the ratio test, this converges absolutely as $0<1$. The answer I have been given states the radius of convergence is $\infty$. Why is this?