Power Series Expansion for Real Arcsine Function

Theorem
The arcsine function has a Taylor Series expansion:


 * $\displaystyle \arcsin x = \sum_{n \mathop = 0}^\infty \frac {\left({2n}\right)!} {2^{2n} \left({n!}\right)^2} \frac {x^{2n + 1}} {2n + 1}$

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

Proof
From Binomial Theorem:

for $-1 < x < 1$.

From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:

We will now prove that the series converges for $-1 \le x \le 1$.

By Stirling's Formula:

By P-Series Converges Absolutely:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^{3/2} }$

is convergent.

So by the Comparison Test, the Taylor series is convergent for $-1 \le x \le 1$.