I am solving via power series the ivp $$y'-2xy=0,\quad y(1)=2.$$ The "solution" is $$y(x)=2\left(1+2(x-1)+3(x-1)^2+\frac{10}{3}(x-1)^3+\frac{19}{6}(x-1)^4+\frac{26}{10}(x-1)^5+\cdots\right)$$ with coefficients generated by the recurrence: $$(n+1)a_{n+1}-2a_{n-1}-2a_n=0,\quad n>0$$ or, $$a_{n+1}=\frac{2(a_n+a_{n-1})}{n+1}.$$ Due to the variable coefficient I'm not sure how to determine the radius of convergence from the recurrence. My naive attempt was to use $$\frac{a_{n+1}}{a_n}=\frac{2\left(1+\frac{a_{n-1}}{a_n}\right)}{n+1}$$ and assume that $$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=R,\quad 0<R<\infty$$ to obtain a contradiction indicating that the radius is either 0 or infinite, but that is not very helpful. The generating function methods don't seem particularly helpful, but I'm quite likely wrong about that. I know that the solution (valid for all $x$) is $$y(x)=2\mathrm{e}^{x^2-1}.$$
Question: How would one obtain the radius of convergence of this series from the recurrence. As an aside are there any general methods that might indicate that the radius is greater than zero?
All $a_n$ are $>0$. From $a_{n+1}\sim{2\over n+1}a_{n-1}$ we see that we gain a factor $\sim{2\over n}$ in two steps. This observation leads to the claim that $$a_n\leq{4^n\over\sqrt{n!}}\qquad(n\geq1)\ .\tag{1}$$ Proof. The statement is true for $n=1$ and $n=2$, by inspection. Assume that it holds for $n-1$ and $n$. Then $$\eqalign{a_{n+1}&\leq{2\over n+1}\left({4^n\over\sqrt{n!}}+{4^{n-1}\over\sqrt{(n-1)!}}\right)\leq {2\over \sqrt{n(n+1)}}\left({4^n\over\sqrt{(n-1)!}}+{4^n\over\sqrt{(n-1)!}}\right)\cr &={4^{n+1}\over\sqrt{(n+1)!}}\ .\cr}$$ Using $(1)$ and, e.g., Stirlings fomula it is then easy to show that $$\rho=\limsup_{n\to\infty}{1\over|a_n|}=\infty\ .$$ As to your second question: The standard proof of Picard's theorem in a complex analysis setting shows that the solution is analytic in a neighborhood of $1\in{\mathbb C}$.