The following question is inspired in the following videos: https://www.youtube.com/playlist?list=PL43B1963F261E6E47
Say one has a general second order linear differential equation $y''+Qy=0$ for some functions $y,Q:U\subseteq\mathbb{R}\rightarrow\mathbb{R}$ with the initial condition $y(0)=a$ and $y'(0)=b$. Replace the equation by $y''+\epsilon Qy=0$ for $\epsilon\in\mathbb{R}$. Assume the solution is of the form $y(x)=\sum_{n=0}^{\infty}y_n(x)\epsilon^n$ for some functions $y_1,\dots,y_n:U\subseteq\mathbb{R}\rightarrow\mathbb{R}$. Then plugging in one finds: $$\sum_{n=0}^{\infty}y''_n\epsilon^n+\sum_{n=0}^{\infty}Qy_n\epsilon^{n+1}=y_0''+\sum_{n=1}^{\infty}\left(y_n''+Qy_{n-1}\right)\epsilon^n=0$$ Therefore, if we let $y_0(x)=a+bx$ and $y_1(0)=\dots=y_n(0)=y_1'(0)=\dots=y_n'(0)=0$ the initial conditions are fulfilled. Finally, one needs to solve $y_n''+Qy_{n-1}=0$ which has the obvious thanks to the initial condition of $y_n(x)=-\int_0^x\int_0^{s_{2n-1}}Q(s_{2n-2})y_{n-1}(s_{2n-2})ds_{2n-2}ds_{2n-1}$. The close form for $y_n$ is $$y_n(x)=(-1)^n\int_0^x\int_0^{s_{2n-1}}Q(s_{2n-2})\int_0^{s_{2n-2}}\int_0^{s_{2n-3}}Q(s_{2n-4})\dots\int_0^{s_{2}}\int_0^{s_1}Q(s_1)(a+s_0b)ds_0\dots ds_{2n-1} \\ =(-1)^n\int_0^x\int_0^{s_{2n-1}}\dots\int_0^{s_1}Q(s_{2n-2})Q(s_{2n-4})\dots Q(s_{0})(a+s_0b)ds_0\dots ds_{2n-1} \\ =(-1)^n\int\limits_{[0,s_1]\times\dots\times[0,s_{2n-1}]\times[0,x]}(a+bs_0)\prod_{\{m\in\mathbb{N}^{\leq2n-2}|\text{m is even}\}}Q(s_m)ds_0\dots ds_{2n-1}$$ and the general solution would be $$y(x)=a+bx+\sum_{n=1}^{\infty}(-1)^n\int\limits_{[0,s_1]\times\dots\times[0,s_{2n-1}]\times[0,x]}(a+bs_0)\prod_{\{m\in\mathbb{N}^{\leq2n-2}|\text{m is even}\}}Q(s_m)ds_0\dots ds_{2n-1}$$ Now, this solution does in fact work for $Q=0$ and $Q=1$ (in the latter the solution yields the taylor expansion of $\sin$ and $\cos$). Non the less, I am guessing that in most cases the solution diverges since otherwise it would be taught in any differential equations class. How does in general one examines the convergence of such a series? In which cases does the solution work? Is there any general rule to assure the convergence of the sequence?