Can convergence of numerical integration methods be related to their accuracy?

Question

I am looking into the convergence of numerical methods of approximation of definite integrals. I plotted the approximation of a specific definite integral vs the number of partitions in the interval for the Trapezoidal rule, Simpson's rule and midpoint rule. All the methods seem to converge at a specific value for number of partitions greater than $1000$. However, some converge faster than the others. So, can I say that the method which converges faster is the most accurate method?