Answer
Diverges for $|x| \gt 1$.
Work Step by Step
$\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 $|x| \gt 1$.
