Math2.org Math Tables: Log Expansions

(Math)

Expansions of the Logarithm Function

Function Summation Expansion Comments

ln (x)

=sum (n=1..inf) (x-1)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)ⁿn

= (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)ⁿn 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) x ⁿ

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

sum (n=1..inf) (-1)ⁿ xⁿ

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

sum (n=1..inf) x^2n-1

2n-1

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

2 (-1 < x < 1)