Power Series Expansion for Real Arccosecant Function
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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
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.32$