How to prove that Taylor series for $\sin^{-1}(x)$ converges at $x=\pm1$

Question

Domain of convergence for $\sin^{-1}(x)$ was asked. I proved that its radius of convergence is $1$. I need to check its convergence at end points. Please tell me some convergence test or some way of proving its convergence at end points.

Answer 1

The series expansion of $\arcsin(x)$ is given by $$\arcsin(x) = \sum_{n=0}^\infty\frac{1}{2^{2n}} \cdot \binom{2n}{n} \cdot \frac{x^{2n+1}}{2n+1} $$ You can use Stirling approximation to show that $\displaystyle \binom{ 2n}{n}\le \frac{4^n}{\sqrt{3n+1}}$ for all $n \ge 1 $ and use $p-$ series test to show convergence.