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, . . . .