Finding the Inverse Function of tanh(x) in the Interval (-1,1)

[annaphys]

Homework Statement

$f:= tanh = \frac{e^x-e^{-x}}{e^x+e^{-x}}$

Prove that
$f^{-1}(x)= \sum\limits_{k=0}^{\infty} \frac{x^{2k+1}}{2k+1}$ for all x in (-1,1)

The Attempt at a Solution

I also found the inverse function to be:

$f^{-1}(x)= \frac{1}{2}ln(\frac{1+x}{1-x})$

I tried working with the taylor polynomial but unfortunately nothing came out of it. Could someone point me in the right direction?

 

[stevendaryl]

Homework Statement

$f:= tanh = \frac{e^x-e^{-x}}{e^x+e^{-x}}$

Prove that
$f^{-1}(x)= \sum\limits_{k=0}^{\infty} \frac{x^{2k+1}}{2k+1}$

The Attempt at a Solution

I also found the inverse function to be:

$f^{-1}(x)= \frac{1}{2}ln(\frac{1+x}{1-x})$

I tried working with the taylor polynomial but unfortunately nothing came out of it. Could someone point me in the right direction?

Do you know the Taylor expansion of $ln(1+x)$? What about $ln(1-x)$? Then you can use a fact about $ln(a/b)$.

 

[annaphys]

Hey thanks for the response! Does that also hold for all x's in (-1,1)? I know that holds for x<<1.