# Encyclopedia of Operations Research and Management Science

2013 Edition
| Editors: Saul I. Gass, Michael C. Fu

# O, o Notation

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-1153-7_200536

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 sufficiently large n, where K is a constant independent of n, then u n = O(v n ); for example, (2n − 1)/(n2 + 1) = O(1/ n). The symbol O (colloquially called “big O”) also extends to the case of functions of a continuous variable; for example, (x + 1) = O(x). O(1) denotes any function that 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 limn−>u n /v n = 0, then u n = o(v n ) (colloquially called “little o”); for example, log n = o(n), where again the notation extends to functions of a continuous variable; for example, sin x = o(x). Furthermore, u n = o(1) means that u n tends to 0 as n tends to infinity; for example, (log n)/n = o(1). In probability modeling (e.g., Markov chains and queueing theory), it is common to see ot) used to represent functions going to 0 faster than a small increment of time Δt, i.e., limΔt−>0[ot)/Δt] = 0.