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
f(t) dt + A + O(f(x)).
To prove this result, compare the areas provided by expressions
f(t) dt. We would like to show that the expression
f(t) dt −
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
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
f(n) + f(M)
f(t) dt f(N) +
f(t) dt − f(n) f(N) − f(M) f(N).
It follows from this that
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.