4 1. Preliminary Notions
Similarly,
I3 = (y y )f(y) f( y + 1)
y
y +1
f(t) dt
= ( y y)f(y)
y +1
y
f(t) dt + f( y + 1).
Since I1 = I I2 I3, it follows that
b
n=a+2
f(n) =
b
a+1
f(t) dt +
x
y
(t t )f (t) dt (x x )f(x)
+
x
b
f(t) dt f( y + 1) ( y y)f(y) +
a+1
y
f(t) dt.
From this, we get that
b
n=a+1
f(n) =
x
y
f(t) dt+
x
y
(t−t )f (t) dt+( x−x)f(x)−( y−y)f(y),
which proves (1.3).
The Euler-MacLaurin formula can be considerably generalized. For in-
stance, one can show the following.
Proposition 1.3. Let ν be a positive integer. Let f be a function such that
the derivatives f , f , . . . , f
(2ν)
are all continuous on the interval [M, N].
Then,
N
n=M
f(n) =
N
M
f(t) dt +
1
2
[f(M) + f(N)] +
B2
2!
f (x)
N
M
+
B4
4!
f (x)
N
M
+ · · · +
B2ν
(2ν)!
f
(2ν−1)(x)
N
M

1
(2ν)!
N
M
B2ν(t t )f
(2ν)(t)
dt,
where the Bi’s are the Bernoulli numbers defined implicitly by

i=0
Bi
yi
i!
=
y
ey 1
and where Bi(x) stands for the i-th Bernoulli polynomial which is defined
as the unique polynomial of degree i satisfying to
x+1
x
Bi(u) du =
xi.
Thus, B1(u) = u
1
2
, B2(u) =
u2
u +
1
6
, B3(u) =
u3

3
2
u2
+
1
2
u, . . . .
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