Maclaurin series for $\operatorname {arccosh}(x)$

Question

Maclaurin series for $\operatorname {arccosh}(x)$

I have asked about the series expansion for $\arccos(x)$here: Maclaurin series for $\arccos(x)$ via integral of $-\dfrac{1}{\sqrt{1-x^2}}$

Is there an identity such as: $\operatorname {arcsinh}(x)+\operatorname {arccosh}(x)=\dfrac{\pi}{2}$?

If the series for $\arccos(x)$ is

$\arccos(x)=\dfrac{\pi}{2}-x-\dfrac{x^3}{6}-\dfrac{3x^5}{40}...$

Then the series for $\operatorname {arccosh}(x)$ is:

$\operatorname {arccosh}(x)= \dfrac{\pi}{2}-x+\dfrac{x^3}{6}-\dfrac{3x^5}{40}...$

Is this a correct guess?

Answer 1

The function $\operatorname{arccosh}$ is not defined at $0$; its domain is $[1,\infty)$. Therefore, it has no MacLaurin series.