Rate of convergence

Question

Suppose that $p_{1},p_{2}>0$ and that $F_{i}(h)=L_{i}+O(h^{p_{i}})$ as $h \to {0^ + }$ for $i=1,2$. What are the rates of convergence of $F_{1}(h)F_{2}(h)$ for various values of $L_{1},L_{2}$?

Answer 1

Just plugging in blindly, if $p = \min(p_1, p_2)$,

$\begin{align} F_1(h) F_2(h) &= (L_1 + O(h^{p_1})(L_2 + O(h^{p_2})) \\ &= L_1 L_2 + L_1 O(h^{p_2}) + L_2 O(h^{p_1}) + O(h^{p_1+p_2}) \\ &= L_1 L_2 + O(h^{p_2}) + O(h^{p_1}) + O(h^{p_1+p_2}) \\ &= L_1 L_2 + O(h^{p}) \\ \end{align} $

This depends on knowing that $O(h^a) + O(h^b) = O(h^a)$ as $h \to 0$ if $0 < a < b$. For example, $O(h)+O(h^2) = O(h)$ since $h^2$ is smaller than $h$ as $h \to 0$.

This, of course, could be considered somewhat abusive, but I enjoy it.