:- Taylor's Formula: If re} has derivative of all orders in an open interval I containing a, then for each positive integer n and for each a: E I, as = as + new a) + Helix a)? + + Mira: mm are} where [More I Hui?!) = Wife\" a) H1 for some c between a and 3:. a The Reminder Estimation Theorem: If there is a positive constant term M such that | f ("+11 {t)| if. M for all t between :L\" and a, inclusive, then the remainder term Rn'E-T} in Taylor's theorem satises the following inequality: '3 _ J|12.+1 {n + 1}! ' If this inequalityr holds for every n and the other conditions of Taylor's theorem are satised by I, then the series converges to at}. lrdmll E M