2 1. Preliminary Notions

One can also obtain a more accurate asymptotic expression in the case

where the function f is decreasing. In fact, one can prove that if f : [1, ∞) →

R+ is continuous, decreasing and such that limx→∞ f(x) = 0, then there

exists a constant A such that

(1.2)

1≤n≤x

f(n) =

x

1

f(t) dt + A + O(f(x)).

To prove this result, compare the areas provided by expressions

∑

1≤n≤x

f(n)

and

x

1

f(t) dt. We would like to show that the expression

D(N) :=

N

1

f(t) dt −

2≤n≤N

f(n),

where N is a positive integer, tends to a positive constant as N → ∞. First,

it is clear that D(N) ≥ 0 for each integer N ≥ 2. So let

R(N) =

∞

n=N+1

n

n−1

f(t) dt − f(n) .

In order to prove (1.2), it is suﬃcient to show that R(N) = O(f(N)). But

for each pair of positive integers M and N with M ≥ N + 3, we have

M−1

n=N+1

f(n) + f(M)

M

N

f(t) dt f(N) +

M−1

n=N+1

f(n),

so that

0

M

n=N+1

n

n−1

f(t) dt − f(n) f(N) − f(M) f(N).

It follows from this that

0 ≤

∞

n=N+1

n

n−1

f(t) dt − f(n) ≤ f(N),

which implies that R(N) = O(f(N)), thereby establishing formula (1.2).

1.2. The Euler-MacLaurin formula

We saw in the previous section that one could approximate the sum of a

function by an integral, provided this function was monotonic. Here, we

will see that if the function has a continuous derivative, then a more precise

approximation can be obtained.