Numerical Analysis - Richardson Extrapolation on Riemann Sum

[Graham87]

Homework Statement

Apply Richardson Extrapolation on a definite integral using Riemann sum.
Then prove that it has the same order of convergence as Trapezoidal rule.

Relevant Equations

Richardson Extrapolation
Riemann Sum

I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?).

For example sin(x) for [0,1], n with trapezoidal rule = 0.420735...
With my own formula I get 0 or much above 0.4207.Cheers!

 

[FactChecker]

Having the same order of convergence does not mean that they will necessarily give the same result. Those are two different things. They should converge to the same value and at the same rate, but may not agree along the way.