O, o Notation

O means “order of” and o means “of lower order than.” If {u _n} and {v _n} are two sequences such that |u _n/v _n| < K for every n greater than some fixed value u _o, where K is a constant independent of n, we write u _n = O(v _n); for example, (2n − 1)/(n ² + 1) = O(1/ n). The symbol O (colloquially called “big O”) is also extended to the case of functions of a continuous variable; for example, (x + 1) = O(x). We denote by O(1) any function x which is defined for all values of x sufficiently large, and which either has a finite limit as x tends to infinity, or at least for all sufficiently large values of x remains less in absolute value than some fixed bound; for example, sin x = O(1).

If the limit of u _n/v _n = 0, we write instead that u _n = o(v _n) (colloquially called “little o”). Thus, u _n < v _n and u _n = o(v _n) are two different ways of expressing the same relation; for example, sin x − x = o(x). The notation is also extended to functions of a continuous variable. Furthermore, u _n = o(1)...