ASL STEM

Definition

Let f be a positive decreasing function defined on [1, infinity) such that the limit as x goes to infinity of f(x) is 0. For all n, define s_n = sum_(k goes from 1 to n)[f(k)], t_n = int_(from 1 to n)[f(x)dx], d_n = s_n - t_n. Then we have: i) 0 < f(n+1) <= d_(n+1) <= d_n <= f(1) for all n. ii) lim d_n exists. iii) sum(f(n)) converges if, and only if, the sequence {t_b} converges. iv) 0<= d_k - lim d_n <= f(k), for all k.

Source: Mathematical Analysis, second edition by Tom M. Apostol