Chapter 10: Infinite Sequences and Series - Section 10.9 - Convergence of Taylor Series - Exercises 10.9 - Page 625: 37

$|x| \lt 0.56968$

Recall the Taylor series for $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ and $\sin x= x-\dfrac{x^3}{6}+\dfrac{ x^5}{120}-....$ We will compute $ x $. $\dfrac{|x|^5}{5!} \lt 5 \times 10^{-4} $ $\implies |x|^5 \lt 5 \times 4 \times 3 \times 2 \times 1 \times 5 \times 10^{-4} $ $\implies |x| \lt 0.56968$