Power Series Expansion for Real Arccotangent Function

Theorem
The arccotangent function has a Taylor series expansion:


 * $\operatorname {arccot} x = \begin{cases} \displaystyle \frac \pi 2 - \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1

\\ \displaystyle \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {\left({2 n + 1}\right) x^{2 n + 1} } & : x \ge 1 \\ \displaystyle \pi + \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {\left({2 n + 1}\right) x^{2 n + 1} } & : x \le 1 \end{cases}$

That is:


 * $\operatorname {arccot} x = \begin{cases} \displaystyle \frac \pi 2 - \left({x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots}\right) & : -1 \le x \le 1

\\ \displaystyle \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \ge 1 \\ \displaystyle \pi + \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \le 1 \end{cases}$

Proof
From Sum of Arctangent and Arccotangent:


 * $\operatorname {arccot} x = \dfrac \pi 2 - \arctan x$

The result follows from Power Series Expansion for Real Arctangent Function.