Definitions
For a given sequence
the transformed sequence is
where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.
for some which often depends on (cf. e.g. Binomial transform). In the simplest case, the and the are real or complex numbers. More generally, they may be elements of some vector space or algebra.
In the context of acceleration of convergence, the transformed sequence is said to converge faster than the original sequence if
where is the limit of, assumed to be convergent. In this case, convergence acceleration is obtained. If the original sequence is divergent, the sequence transformation acts as extrapolation method to the antilimit .
If the mapping is linear in each of its arguments, i.e., for
for some constants (which may depend on n), the sequence transformation is called a linear sequence transformation. Sequence transformations that are not linear are called nonlinear sequence transformations.
Read more about this topic: Sequence Transformation
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