parameter ▁group , ▁a ▁foundation ▁for ▁physical ▁models . ▁ ▁Series ▁expansion ▁( vel ocity ) ▁The ▁Lor ent z ▁factor ▁has ▁the ▁Mac la ur in ▁series : ▁ ▁which ▁is ▁a ▁special ▁case ▁of ▁a ▁bin omial ▁series . ▁ ▁The ▁approximation ▁▁ ≈ ▁ 1 ▁+ ▁▁ β 2 ▁may ▁be ▁used ▁to ▁calculate ▁relativ istic ▁effects ▁at ▁low ▁spe eds . ▁It ▁holds ▁to ▁within ▁ 1 % ▁error ▁for ▁v ▁< ▁ 0 . 4 ▁c ▁( v ▁< ▁ 1 2 0 , 0 0 0   km / s ), ▁and ▁to ▁within ▁ 0 . 1 % ▁error ▁for ▁v ▁< ▁ 0 . 2 2   c ▁( v ▁< ▁ 6 6 , 0 0 0   km / s ). ▁ ▁The ▁trunc ated ▁versions ▁of ▁this ▁series ▁also ▁allow ▁phys ic ists ▁to ▁prove ▁that ▁special ▁relativ ity ▁reduces ▁to ▁Newton ian ▁mechan ics ▁at ▁low ▁spe eds . ▁For ▁example , ▁in ▁special ▁relativ ity , ▁the ▁following ▁two ▁equations ▁hold : ▁ ▁For ▁▁ ≈ ▁ 1 ▁and ▁▁ ≈ ▁ 1 ▁+ ▁▁ β 2 , ▁respectively , ▁these ▁reduce ▁to ▁their ▁Newton ian ▁equival ents : ▁ ▁The ▁Lor ent z ▁factor ▁equation ▁can ▁also ▁be ▁invert ed ▁to ▁yield ▁ ▁This ▁has ▁an ▁asympt otic ▁form ▁ ▁. ▁ ▁The ▁first ▁two ▁terms ▁are ▁occasionally ▁used ▁to ▁quickly ▁calculate ▁veloc ities ▁from ▁large ▁ ▁values . ▁The ▁approximation ▁ β ▁ ≈ ▁ 1 ▁− ▁