section_path: Properties/Order of convergence
section_name: Order of convergence
paragraph_text: Order of convergence
  
  Beyond whether or not a sequence { a n } {\displaystyle \{a_{n}\}} converges to a limit a {\displaystyle a}, it is possible to describe how fast a sequence converges to a limit. One way to quantify this is using the order of convergence of a sequence.
  
  A formal definition of order of convergence can be stated as follows. Suppose { a n } n > 0 {\displaystyle \{a_{n}\}_{n>0}} is a sequence of real numbers which is convergent with limit a {\displaystyle a}. Furthermore, a n ≠ a {\displaystyle a_{n}\neq a} for all n {\displaystyle n}. If positive constants λ {\displaystyle \lambda } and α {\displaystyle \alpha } exist such that lim n → ∞ | a n + 1 − a | | a n − a | α = λ {\displaystyle \lim _{n\to \infty }{\frac {\left|a_{n+1}-a\right|}{\left|a_{n}-a\right|^{\alpha }}}=\lambda } then a n {\displaystyle a_{n}} is said to converge to a {\displaystyle a} with order of convergence α {\displaystyle \alpha }. The constant λ {\displaystyle \lambda } is known as the asymptotic error constant.
  
  Order of convergence is used for example the field of numerical analysis, in error analysis.