Theorem 13

Given the point closed Newton-Cotes formula. Then there exists a in so that

,

for even, when has continuous derivatives up to the -th one over ;

and

for odd, when has continuous derivatives up to the -th one over ;

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Notice that the formula using an odd number of points ( even) is about as accurate than the formula with one more point. Therefore, one prefers to use a formula with odd number of points and increase points in pairs to increase the accuracy!

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Since the Newton-Cotes formula with points is based upon the use of the Lagrange Polynomial through these points, which is of degree , the formula should be correct when is a polynomial with degree smaller or equal to . Therefore, the error term should disappear for polynomials up to degree so that we expect the term in to be present. Notice that for an odd number of points used, the formula achieves a better result!

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