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Why 1/n^2?


When computing exact integrals, we have an infinite number of intervals and h is infinitely small (very close to 0), so if h is the length of the intervals, why wouldn't the error be O(h) or O(h*#number of intervals)?


O(1/n^2) is a big-O expression I've never really seen before - how does it match up to other big-O expressions in terms of "speed"?


I wonder if there are pathological examples that are continuous, but perhaps not even differentiable, where the trapezoidal rule can be made to be arbitrarily bad? What would these look like?