Here is what I thought but two formulas can't match:
1.we know this formula below Richardson Extrapolation
2.we know these two formulas for composite Simpson's rule Composite Simpson's Rule
3.we know that $T_{0,m}$ is the value of the composite trapezoid rule approximation where the subintervals have length h, and $T_{0,m+1}$ is the value where the subintervals have length $\frac{h}{2}$
4.Therefore, plug in what we have in step3 into step2, then use step1 formula to calculate $T_{1,m}$ see my calculation
I figured out that there is still one term $$4\sum_{j=0}^{N-1}f(\frac{x_{j}+x_{j+1}}{2})$$missing in the parenthesis compared to the formula. Is there anything wrong with my process? Appreciate any help!!!!