The FixedPointList command returns a list of results from applying a function until the results no longer change.

$\mathrm{with}\left(\mathrm{MmaTranslator}\left[\mathrm{Mma}\right]\right)\:$

$\mathrm{FixedPointList}\left(\cos \,0.5\right)$

$\left[0.5\,0.6390124942\,0.8026851007\,0.6947780268\,0.7681958313\,0.7191654459\,0.7523557594\,0.7300810632\,0.7451203413\,0.7350063090\,0.7418265227\,0.7372357254\,0.7403296519\,0.7382462383\,0.7396499628\,0.7387045393\,0.7393414523\,0.7389124493\,0.7392014442\,0.7390067797\,0.7391379108\,0.7390495806\,0.7391090814\,0.7390690012\,0.7390959998\,0.7390778133\,0.7390900640\,0.7390818118\,0.7390873706\,0.7390836261\,0.7390861484\,0.7390844494\,0.7390855938\,0.7390848230\,0.7390853422\,0.7390849924\,0.7390852281\,0.7390850693\,0.7390851763\,0.7390851042\,0.7390851528\,0.7390851200\,0.7390851421\,0.7390851272\,0.7390851373\,0.7390851305\,0.7390851350\,0.7390851320\,0.7390851340\,0.7390851327\,0.7390851336\,0.7390851330\,0.7390851334\,0.7390851331\,0.7390851333\,0.7390851332\,0.7390851332\right]$

$\mathrm{with}\left(\mathrm{MmaTranslator}\right)\:$

$\mathrm{FromMma}\left(\mathrm{`FixedPointList[Cos,\; 0.5]`}\,\mathrm{evaluate}\right)$

$\left[0.5\,0.6390124942\,0.8026851007\,0.6947780268\,0.7681958313\,0.7191654459\,0.7523557594\,0.7300810632\,0.7451203413\,0.7350063090\,0.7418265227\,0.7372357254\,0.7403296519\,0.7382462383\,0.7396499628\,0.7387045393\,0.7393414523\,0.7389124493\,0.7392014442\,0.7390067797\,0.7391379108\,0.7390495806\,0.7391090814\,0.7390690012\,0.7390959998\,0.7390778133\,0.7390900640\,0.7390818118\,0.7390873706\,0.7390836261\,0.7390861484\,0.7390844494\,0.7390855938\,0.7390848230\,0.7390853422\,0.7390849924\,0.7390852281\,0.7390850693\,0.7390851763\,0.7390851042\,0.7390851528\,0.7390851200\,0.7390851421\,0.7390851272\,0.7390851373\,0.7390851305\,0.7390851350\,0.7390851320\,0.7390851340\,0.7390851327\,0.7390851336\,0.7390851330\,0.7390851334\,0.7390851331\,0.7390851333\,0.7390851332\,0.7390851332\right]$