Trapezoid rule to Simpson's by Richardson extrapolation

Hi,

**problem:**

The trapezoid formula (for one interval) is given by,

a) The trapezoid method has a global error of . In the above equation we use . Simpson's 1/3-rule can be derived by performing Richardson extrapolation on the trapoezoid formula. Do this.

b) What happens with the error?

**attempt:**

First of all I divide the interval into two intervals,

This approximation has global error and so I am tempted to write that,

but I've read that it can be shown that for the composite trapezoid formula we have,

The book I have does not show how it can be shown, it merely points out that it cannot be shown very easily..I will go with it and use that result for now.

I let to get,

Here's the part that kind of confuses me. The trapezoid rule has local error . I am then tempted to say that,

,

but then I do not see how Richardson extrapolation can do any good...

If I instead use the global error, I can perform Richardson extrapolation and get,

b) The global error is .

Like usual, I write too much to say too little!

Thanks.