For a given [[Quadrature]] method $A(h)$ (where $h$ is the step size) $\large \begin{align} A(h) &= A+ k_{n}h^{n} + \mathcal{O}(h^{{n+2}})\\ A(2h) &= A + 2^{n}k_{n}h^{n} + \mathcal{O}(h^{n+2}) \end{align} $ We can use this as a [[System of Linear Equations]] as the big $O$ term is negligible. $\large \begin{align} A(2h) - A(h) &= (2^{n}-1)k_{n}h^{n} + \mathcal{O}\pa{h^{n+2}} \\ k_{n}h^{n} &= \frac{A(2h)-A(h)}{2^{n}-1} + \mathcal{O} \left(h^{n+2}\right) \end{align} $ $\large \begin{align} A(h) &= A+ \frac{A(2h)-A(h)}{2^{n}-1} + \mathcal{O} \left(h^{n+2}\right) \\ &= A(h) - \frac{A(2h)-A(h)}{2^{n}-1} + \mathcal{O}(h^{n+2}) \end{align}$