*O, o* Notation

**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, (2*n* − 1)/(*n*^{2} + 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 lim_{n−>}_{∞}*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 *o*(Δ*t*) used to represent functions going to 0 faster than a small increment of time Δ*t*, i.e., lim_{Δt−>}_{0}[*o*(Δ*t*)/Δ*t*] = 0.