Power Series Expansion for Real Arccotangent Function

Theorem
The arccotangent function has a Taylor series expansion:


 * $\arccot x = \begin {cases} \ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1

\\ \ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \ds \pi + \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \le -1 \end {cases}$

That is:


 * $\arccot x = \begin {cases} \ds \frac \pi 2 - \paren {x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots} & : -1 \le x \le 1

\\ \ds \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \ge 1 \\ \ds \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:


 * $\arccot x = \dfrac \pi 2 - \arctan x$

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