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} \dfrac \pi 2 - \paren {x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \cdots} & : -1 \le x \le 1 \\

\dfrac 1 x - \dfrac 1 {3 x^3} + \dfrac 1 {5 x^5} - \cdots & : x \ge 1 \\ \pi + \dfrac 1 x - \dfrac 1 {3 x^3} + \dfrac 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.

$\blacksquare$

Also see

• Power Series Expansion for Real Arcsine Function
• Power Series Expansion for Real Arccosine Function
• Power Series Expansion for Real Arctangent Function
• Power Series Expansion for Real Arcsecant Function
• Power Series Expansion for Real Arccosecant Function

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.30$
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions