Suppose I have an infinite series, $$0= a_0 x^p + a_1 x^{p+1} + a_2 x^{p+2} + \mathcal{O}(x^{p+3})$$ where $x<1$ and $\mathbb{N}\ni p\geq 1$.
Under what conditions does it hold that $a_0=0$? In a proof, it was stated that $a_0=0$ because terms of order $\mathcal{O}(x^{p+1})$ cannot correct for this, otherwise i.e. $a_0 \neq 0$ would imply the series does not converge to $0$.
I don't immediately see why. Maybe it is not true in general, but under certain conditions?!
The problem is oddly stated. Where is the infinite series? I see four terms added together. Nevertheless, we can show $a_0=0.$ On the interval $(-1,1)$ we are given
$$a_0x^{p}+a_1x^{p+1}+a_2x^{p+2} + O(x^{p+3})=0$$
(where you should say $O(x^{p+3})$ as $x\to 0$). Rewrite the above as
$$x^p(a_0+ a_1x+a_2x^{2} + O(x^{3}))=0.$$
For $x\ne0,$ we can divide by $x^p$ to get
$$a_0+ a_1x+a_2x^{2} + O(x^{3})=0.$$
Take the limit of this as $x\to 0.$ On the left we get $a_0.$ On the right we get $0.$ Therefore $a_0=0.$