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.
