In reply to Rick Wicklin.

Thanks for your reply. I did appreciate the different convergence rates and just guessed that p = 1/3 was about right. My amazement was really how much better the approximation becomes. For example consider the errors for n = 449, A_in = 1E-4, A_out = 5E-5 and A_WgtAv = 1E-9. I am thinking some more about your challenge.