1

I've got the following formula for a homework programming task.

$$\ln(x) = 2\sum_{k = 0}^{\infty}\frac{(x-1)^{2k+1}}{(2k+1)(x+1)^{2k+1}}$$

I was wondering what the name of the formula is, or where it comes from? After searching on the internet I couldn't find it.

Another think I'd like to know is if there are other $\ln(x)$ approximation formulas that approach $\ln(x)$ faster?

Simply Beautiful Art
  • 74,685
akuzminykh
  • 145

1 Answers1

1

The formula you gave came from the series expansion for the $\textbf{Inverse Hyperbolic Tangent}$ function :

$$ \left(\forall x\in\left]-1,1\right[\right),\ \tanh^{-1}{x}=\sum_{n=0}^{+\infty}{\frac{x^{2n+1}}{2n+1}} $$

and the fact that $ \left(\forall x\in\left]-1,1\right[\right),\ \tanh^{-1}{x}=\frac{1}{2}\ln{\left(\frac{1+x}{1-x}\right)} \cdot $

CHAMSI
  • 8,333