## Thomas' Calculus 13th Edition

Diverges for $|x| \gt 1$.
$\lim\limits_{n \to \infty}|\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} |\dfrac{(-1)x^2 (2n-1)}{2n+1}|$ or, $=x^2 \lim\limits_{n \to \infty} \dfrac{2-1/n1}{2+1/n}$ or, $=x^2$ Now, $| Error|=|\dfrac{(-1)^n x^n}{n}|=\dfrac{1}{n (10^n)}$ But $\dfrac{1}{2n-1} \lt 10^{-3}$ or, $\dfrac{1}{2n-1} \lt \dfrac{1}{10^{3}}$ or, $n=501$ Thus, the series is divergent for $|x| \gt 1$.