Let s be the sum of a series ∑ an that has been shown to be convergent by the Integral Test and let f(x) be the function in that test. The remainder after n terms is
Thus Rn is the error made when sn, the sum of the first n terms, is used as an approximation to the total sum s.
- (a) By comparing areas in a diagram like Figures 3 and 4 (but with x … n), show that
- (b) Deduce from part (a) that
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Essential Calculus
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