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 Log Expansions (Math | Calculus | Series | Log)

Expansions of the Logarithm Function
Function Summation Expansion Comments
ln (x)
 = (x-1)n  n
= (x-1) - (1/2)(x-1)2 + (1/3)(x-1)3 + (1/4)(x-1)4 + ...
Taylor Series Centered at 1
(0 < x <=2)
ln (x)
 = ((x-1) / x)n  n
= (x-1)/x + (1/2) ((x-1) / x)2 + (1/3) ((x-1) / x)3 + (1/4) ((x-1) / x)4 + ...

(x > 1/2)
ln (x)
 =ln(a)+ (-1)n-1(x-a)n  n an
= ln(a) + (x-a) / a - (x-a)2 / 2a2 + (x-a)3 / 3a3 - (x-a)4 / 4a4 + ...
Taylor Series
(0 < x <= 2a)
ln (x)
 =2 ((x-1)/(x+1))(2n-1)  (2n-1)
= 2 [ (x-1)/(x+1)  + (1/3)( (x-1)/(x+1) )3 + (1/5) ( (x-1)/(x+1) )5 + (1/7) ( (x-1)/(x+1) )7 + ... ]
(x > 0)

Expansions Which Have Logarithm-Based Equivalents
Summation Expansion Equivalent Value Comments
 x n  n
= x + (1/2)x2  +(1/3)x3 + (1/4)x4 + ...
= - ln (x + 1) (-1 < x <= 1)
 (-1)n xn  n
= - x + (1/2)x2 - (1/3)x3 + (1/4)x4 + ...
= - ln(x) (-1 < x <= 1)
 x2n-1  2n-1
= x + (1/3)x3 + (1/ 5)x5 + (1/7)x7 + ...
= ln ( (1+x)/(1-x) )
2
(-1 < x < 1)