Answer
$$n \geq 4$$
Work Step by Step
Consider: $S_n=a_1-a_2+......+(-1)^{n+1}a_n$
and$$|S-S_n| \leq |a_{n+1}| \\ \implies |a_{n+1} | \lt 0.001 \\ \implies |\dfrac{1}{[(n+1)+3 \sqrt {n+1}]^3}| \lt 0.001 \\ ((n+1)+3 \sqrt {n+1})^3 \gt 1000 \\ \implies n \geq 4$$
