Answer
True
Work Step by Step
If $|\lim\limits_{n \to \infty}a_n|>0$ or is undefined, the series diverges.
If $|\lim\limits_{n \to \infty}a_n|=0$, the limit test is inconclusive.
Now $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty} \frac{n}{1000 (n + 1)}=\frac{1}{1000}\neq0$
Since the limit is not $0$, the series $\sum_n^∞ \frac{n}{1000 (n + 1)}$ diverges.
