Euler-Maclaurin sum formula

The Euler-Maclaurin integration and sums formulas can be derived from Darboux's formula by substituting the Bernoulli polynomial B_n(t) in for the function ϕ(t). Differentiating the identity B_n(t + 1) - B_n(t) = n t^(n - 1) n - k times gives B_n^(n - k)(t + 1) - B_n^(n - k)(t) = n(n - 1)...k t^(k - 1). Plugging in t = 0 gives B_n^(n - k)(1) = B_n^(n - k)(0).

Darboux's formula | Maclaurin-Cauchy theorem | sum | Wynn's epsilon method