Editing Rate of convergence (section)

=== Accelerating convergence rates ===
{{Main|Series acceleration}}
Many methods exist to accelerate the convergence of a given sequence, i.e., to [[sequence transformation|transform one sequence]] into a second sequence that converges more quickly to the same limit. Such techniques are in general known as "[[series acceleration]]" methods. These may reduce the [[computational cost|computational costs]] of approximating the limits of the original sequences. One example of series acceleration by sequence transformation is [[Aitken's delta-squared process]]. These methods in general, and in particular Aitken's method, do not typically increase the order of convergence and thus they are useful only if initially the convergence is not faster than linear: if <math>(x_k)</math> converges linearly, Aitken's method transforms it into a sequence <math>(a_k)</math> that still converges linearly (except for pathologically designed special cases), but faster in the sense that <math display="inline">\lim_{k \rightarrow \infty} (a_k-L)/(x_k-L)= 0</math>. On the other hand, if the convergence is already of order &ge; 2, Aitken's method will bring no improvement.