I'm stuck on a question regarding numerical methods.
The aim of this question is to approximate $ f''(x) + f'(x) $ with the highest possible order of truncation error using $Q(h) = \frac{\alpha_{-1} f(x-h) + \alpha_0 f(x) + \alpha_1 f(x+h)}{h^2}$.
The first part of the question asks for proof that $\alpha_{-1} = 1 - \frac{h}{2}$, $\alpha_0 = -2$, $\alpha_1 = 1 + \frac{h}{2}$. This is quite straightforward using Taylor series; however, in the solution the Taylor polynomials are calculated up until a remainder term of $\mathcal{O}(h^5)$.
My question is, why specifically $\mathcal{O}(h^5)$? The terms with the 3rd and 4th derivative are not necessary for calculating the $\alpha_j$ values, and if the aim is a highest possible truncation error, why stop at $\mathcal{O}(h^5)$?
Thanks in advance,
L. Roelandschap