Question: Suppose that N(h) is an approximation to $M$ for every $h > 0$ and that $M = N(h) + K_1 h + K_2 h^2 + K_3 h^3 +\cdots$, for some constants $K_1, K_2, K_3,\cdots$. Use the values $N(h), N( h/3),$ and $N (h/9)$ to produce an $O(h^3)$ approximation to $M$.
My work:
$$M=N(h)+K_1 h+K_2 h^2 + K_3 h^3+\cdots$$ $$M=N(\frac{h}{3})+\frac{K_1}{3} h+\frac{K_2}{9}h^2 + \frac{K_3}{27}h^3+\cdots$$ $$M=N(\frac{h}{9})+\frac{K_1}{9}h+\frac{K_2}{81}h^2 + \frac{K_3}{729}h^3+\cdots$$
I'm not sure what to do next.
I was told that $N_1(h)$ gives $O(h^2)$, $N_2(h)$ gives $O(h^4)$, $N_3(h)$ gives $O(h^6)$, $N_4(h)$ gives $O(h^8)$, so I'm not sure how to get $O(h^3)$.