 Provided the series converges, formula_3 will also approach the limit formula_7 as formula_8
The Shanks transformation formula_9 of the sequence formula_3 is the new sequence defined by

where this sequence formula_9 often converges more rapidly than the sequence formula_13 
Further speed-up may be obtained by repeated use of the Shanks transformation, by computing formula_14 formula_15 etc.Note that the non-linear transformation as used in the Shanks transformation is essentially the same as used in Aitken's delta-squared process so that as with Aitken's method, the right-most expression in formula_9's definition (i.e.formula_17) is more numerically stable than the expression to its left (i.e.