(Using an integral to improve convergence) Recall the error
Question and Solution
(Using an integral to improve convergence) Recall the error formula for the Midpoint Rule, according to which Z kC1=2 k?
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(Using an integral to improve convergence) Recall the error formula for the Midpoint Rule, according to which Z kC1=2 k?1=2 f .x/ dx ? f .k/ D f 00.c/ 24 ; where k ? .1=2/ c k C .1=2/. (a) If f 00.x/ is a decreasing function of x, show that f 0 .k C 3 2 / ? f 0 .k C 1 2 / f 00.c/ f 0 .k ? 1 2 / ? f 0 .k ? 3 2 /: (b) If (i) f 00.x/ is a decreasing function of x, (ii) R 1 NC1=2 f .x/ dx converges, and (iii) f 0 .x/ ! 0 as x ! 1, show that f 0 .N ? 1 2 / 24 X1 nDNC1 f .n/ ? Z 1 NC1=2 f .x/ dx f 0 .N C 3 2 / 24 : (c) Use the result of part (b) to approximate P1 nD1 1=n2 to within 0.001.
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