Themes > Science > Mathematics > Calculus > Series Expansions > Log Expansions 

Expansions of the Logarithm Function
Function Summation Expansion Comments
  ln (x)
=sum (n=1..inf) (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)
=sum (n=1..inf) ((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)+sum (n=1..inf)  (-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)
=2sum (n=1..inf) ((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
Summantion Expansion Equivalent Value Comments
sum (n=1..inf) (-1)n xn 

n

= x + (1/2)x2  +(1/3)x3 + (1/4)x4 + ...

 = - ln (x + 1) (-1 < x <= 1)

 
sum (n=1..inf) x2n-1 

2n-1

= x + (1/3)x3 + (1/ 5)x5 + (1/7)x7 + ...

 = ln ( (1+x)/(1-x) ) 

2

 

 (-1 < x < 1)


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