Power Series Expansion for Real Arccosine Function

Theorem

The arccosine function has a Taylor Series expansion:

$\ds \arccos x$

$=$

$\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$

$\ds $

$=$

$\ds \frac \pi 2 - \paren {x + \frac {x^3} {2 \times 3} + \frac {\paren {1 \times 3} x^5} {2 \times 4 \times 5} + \frac {\paren {1 \times 3 \times 5} x^7} {2 \times 4 \times 6 \times 7} + \cdots}$

which converges for $-1 \le x \le 1$.

Proof

$\ds \arccos x$

$=$

$\ds \frac {\pi} 2 - \arcsin x$

Sum of Arcsine and Arccosine

$\ds $

$=$

$\ds \frac {\pi} 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$

Power Series Expansion for Real Arcsine Function

It follows from Power Series Expansion for Real Arcsine Function that the series is convergent for $-1 \le x \le 1$.

$\blacksquare$

Also see

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.28$
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
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.28.$