Dave's Math Tables: Root Expansions

(Math | Calculus | Expansions | Series | Root)

Expansions of the Root Functions

Function	Expansion	Summation
(x)  (0<=x<=2)	= [1] (1/0!) + [1/2] (1/1!) (x-1) + [(1/2) (-1/2)] (1/2!) (x-1)2 + [(1/2) (-1/2) (-3/2)] (1/3!) (x-1)3 + [(1/2) (-1/2) (-3/2) (-5/2)] (1/4!) (x-1)4 + ...    = 1 + (1/2)(x-1) - (1/8)(x-1)2 + (1/16)(x-1)3 - (5/128)(x-1)4 + ...	Very akwardly expressed:  1+ (-1)n-1 n (2n-2)! (x-1)n 22n-1 (n!)2 or:  (3/2) (x-1)n  (3/2 - n) n!
1 / (x) (0<x<=2)	= [1] (1/0!) + [-1/2] (1/1!) (x-1)1 + [(-1/2)(-3/2)] (1/2!) (x-1)2 + [(-1/2)(-3/2)(-5/2) (1/3!) (x-1)3 + ... = 1 - (1/2)(x-1) + (3/8)(x-1)2 - (5/16)(x-1)3 + (35/128)(x-1)4 -  ...	Very akwardly expressed:  (-1)n (2n)! (x-1)n  22n (n!)2 or:  (1/2) (x-1)n  (1/2 - n) n!