Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that:

Where, $T(n) =$ trapezoidal rule formula

and have gathered that $$T(2n) = {1\over 2} T(n) + \text{sum of the odd terms}$$

From this, is it right to say $T(n)$ is a more accurate calculation than $T(2n)$, right?

Another question I have is that I have a question in which I have to calculate $S(2n)$ using richardson extrapolation which i have the formula:

$$S(2n) = {4\over 3} T(2n) - {1\over 3} T(n)$$ From this, is it right to conclude that $S(n)$ is more accurate than $S(2n)$?

....I have a few more questions to follow up on, but for now I just wanted to check my understanding

If that is all correct, I was wondering how do we actually start calculating $S(2n)$ and how to find $S(4n)$ and to express $S(4n)$ in terms of $S(2n)$ without $T(2n)$ or $T(n)$ in the formula

• I reformatted your question a bit. Please check to make sure that I didn't change what you are trying to ask. You can find more information about formatting here: meta.math.stackexchange.com/questions/5020/…. Also, if you have more questions, then remember to post them as separate questions. It is best to post separate questions as separate questions. – Thomas May 9 '13 at 15:48
• Thank you for the formatting and link! :) I've been looking for something like for ages!!! :) – orangesun May 9 '13 at 16:07
• Glad to help. Also here: meta.math.stackexchange.com/questions/370/formatting-sandbox you can try to experiment with formatting. – Thomas May 9 '13 at 16:12