1.2. The Euler-MacLaurin formula 3 Proposition 1.2. (Euler-MacLaurin formula) Let 0 y x be two real numbers and assume that f : [y, x] → R has a continuous derivative on [y, x]. Then (1.3) yn≤x f(n) = x y f(t) dt+ x y (t−t )f (t) dt+( x−x)f(x)−( y−y)f(y). Proof. Expanding the second integral on the right side of (1.3), we get I = x y (t − t )f (t) dt = x y +1 + x x + y +1 y (t − t )f (t) dt = I1 + I2 + I3, say. Set a = y and b = x , and let us evaluate separately each of the integrals Ii. I1 = b a+1 (t − t )f (t) dt = b−1 k=a+1 k+1 k (t − k)f (t) dt = b−1 k=a+1 k+1 k tf (t) dt − b−1 k=a+1 k k+1 k f (t) dt = b−1 k=a+1 tf(t) t=k+1 t=k − k+1 k f(t) dt − b−1 k=a+1 k(f(k + 1) − f(k)) = bf(b) − (a + 1)f(a + 1) − b a+1 f(t) dt + (a + 1)f(a + 1) − bf(b) + b n=a+2 f(n). Therefore, b n=a+2 f(n) = b a+1 f(t) dt + I1. On the other hand, I2 = x b (t − b)f (t) dt = x b tf (t) dt − b x b f (t) dt = tf(t) t=x t=b − x b f(t) dt − b x b f (t) dt = (x − x )f(x) − x x f(t) dt.
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