FixedPointList

FixedPointList[f, expr] generates a list of results from applying f repeatedly until the result no longer changes.

FixedPointList[f, expr, n] stops after at most n steps.

Examples

Find fixed point iterations:

FixedPointList[Cos, 1.0, 10]
(* {1., 0.540302, 0.857553, 0.65429, 0.79348, ...} *)

Newton's method for sqrt(2):

FixedPointList[(# + 2/#)/2 &, 1.0]
(* {1., 1.5, 1.41667, 1.41422, 1.41421, 1.41421, 1.41421} *)

Please visit the official Wolfram Language Reference for more details.