Here is the formula for determining order of convergence, $q$ is the order of convergence if we can find a constant $\mu$ that the fraction converges to as $k \to \inf$...
$\lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = \mu \,\big|\; \mu > 0.$
I know that this formula applies to root finding methods such as Newtons method or the Secant method where we can always take another iteration and find a value closer to the root. So we can let the number of iterations taken, $k$, approach infinity.
But does this formula apply when it comes to initial value problems and we are using Eulers method or Runge Kutta methods? As we set a fixed number of iterations dependent on the step size and the location $x$ where we want to find the value of the function. So if $x_0 = 0$, we want to find the value at $x = 4$ and we have a step size of $0.5$ we will have $k = 8$ iterations. We can't just 'take another iteration' and find a value closer, the number of iterations are set.
So in that case does the formula above not apply to initial value problems?