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Root Expansions 
(Math | Calculus | 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 awkwardly expressed: 
1+sum (n=1..inf) (-1)n-1 n (2n-2)! (x-1)n
22n-1 (n!)2
or: 
sum (n=0..inf) Gamma(3/2) (x-1)
Gamma
(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 awkwardly expressed: 
sum (n=0..inf) (-1)n (2n)! (x-1)
22n (n!)2
or: 
sum (n=0..inf) Gamma(1/2) (x-1)
Gamma
(1/2 - n) n!
 

  
 
  

 
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