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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