Power Series Expansion for Real Arccosecant Function

Theorem

The arccosecant function has a Taylor Series expansion:

\(\ds \arccsc x\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} x^{2 n + 1} }\)

\(\ds \)

\(=\)

\(\ds \frac 1 x + \frac 1 {2 \times 3 x^3} + \frac {1 \times 3} {2 \times 4 \times 5 x^5} + \frac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 7 x^7} + \cdots\)

which converges for $\size x \ge 1$.

Proof

From Arcsine of Reciprocal equals Arccosecant:

$\arccsc x = \arcsin \dfrac 1 x$

From Power Series Expansion for Real Arcsine Function:

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

which is converges for $\size x \le 1$.

The result follows by subtituting $\dfrac 1 x$ for $x$.

This converges for $\size {\dfrac 1 x} \le 1$, that is, for $\size x \ge 1$.

$\blacksquare$

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 Arccotangent Function

• Power Series Expansion for Real Arcsecant Function

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.32$