Euler–Maclaurin Summation
Build a finite sum from its integral, endpoint and derivative corrections and see the correction that makes it exact.
Free theory articleBuild a finite sum from its integral, endpoint and derivative corrections and see the correction that makes it exact.
Concept & interpretation
The Euler–Maclaurin summation formula expresses a finite discrete sum Σk=1N f(k) as the continuous definite integral ∫0N f(x) dx plus half-endpoint and Bernoulli odd-derivative boundary corrections:
Interactive Experiment
Explore how polynomial degree determines which Bernoulli correction closes the gap in the Euler–Maclaurin Summation interactive workspace:
- With Power = Square, Upper N = 4, and Correction = B2, verify that Integral term = 21.333333333333332, Endpoint term = 8, and B2 term = 0.6666666666666666 sum to Selected estimate = 30 (error
0). - Set Power = Cube, Upper N = 2, and Correction = Endpoints. Note that
4 + 4 = 8misses Exact finite sum = 9 by the omitted B2 term = 1. - Switch Power = Fourth power with Upper N = 2 and Correction = B2. Observe that
32/5 + 8 + 8/3 = 256/15 ≈ 17.066666666666666overshoots17by1/15becausef'''(x) = 24xrequires the negative B4 term = -1/15. - Change Correction to B4 on Fourth power (with Upper N = 1 or 2) to see the negative fourth-order Bernoulli term restore exactness.