Answer
$$n \geq 999$$
Work Step by Step
Consider: $S_n=a_1-a_2+......+(-1)^{n+1}a_n$
and $$|S-S_n| \leq |a_{n+1}| \\ \implies |a_{n+1} | \lt 0.001 =|\dfrac{n+1}{(n+1)^2+1}| \lt 0.001 \\ \implies (n+1)^2+1 \gt 1000(n+1) \\ \implies n \geq 999$$
