Applying the initial conditions on a Power Series

Question

Consider the IVP: \begin{equation} y''+y'=xy\quad;\quad y(0)=2,y'(0)=1 \end{equation} Using the Power Series method to get the solution, we'll initially write: \begin{equation} y=\sum_{n=0}^{\infty}a_nx^n\quad;\quad y'=\sum_{n=1}^{\infty}na_nx^{n-1} \end{equation} Applying the initial conditions, one should get $a_0=2$ and $a_1=1$. And that's where I have a little conceitual confusion. The argument I would use to illustrate that would simply be: \begin{equation} y=a_0+\sum_{n=1}^{\infty}a_nx^n\Longrightarrow y(0)=a_0=2\quad;\quad y'=a_1+\sum_{n=2}^{\infty}na_nx^{n-1}\Longrightarrow y'(0)=a_1=1 \end{equation} Since the other terms will clearly vanish. That's a decent approach, but it seems to me to be more of an adjustment than a proper argument. If we, without doing the previous development, tried to evaluate $y(0)$ and $y'(0)$, the other therms would continue to vanish, but we'd get $0^0$ forms at $n=0$. Well, that must be a way to argue that this continues to give $a_0$ and $a_1$ regardless. I thought about using the fact: \begin{equation} \lim\limits_{t\rightarrow 0}t^0=1 \end{equation} But it seems like a rather tautological argument. How can I formally develop a argument for that?