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.
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