<s>[INST] Use infinite series to prove Use infinite series to prove that
$$\arcsin{x}\lt \frac{x}{1-x^2},$$ for $0\lt x\lt1$.
 [/INST] Let $f(x)=\frac{x}{1-x^2}-\arcsin{x}$.
Hence, $$f'(x)=\frac{1-x^2-x(-2x)}{(1-x^2)^2}-\frac{1}{\sqrt{1-x^2}}=\frac{1+x^2-\sqrt{(1-x^2)^3}}{(1-x^2)^2}=$$
$$=\frac{ x^2(x^4-2x^2+5)}{(1+x^2-\sqrt{(1-x^2)^3})(1-x^2)^2}>0.$$
Thus, $f(x)>f(0)=0$ and we are done!
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