Chapter 9 - Section 9.6 - Alternating Series and Conditional Convergence - Exercises - Page 521: 53

$n \geq 31$

We have $S_n=a_1-a_2+......+(-1)^{n+1}a_n$ Now, $|S-S_n| \leq |a_{n+1}|$ So, $|\space Error | \lt 0.001 =|\dfrac{1}{(n+1)^2+3}| \lt 0.001$ or, $(n+1)^2+3 \gt 1000$ or, $n \geq 31$ Thus, we need $n \geq 31$ terms to estimate the sum of the entire series with an error is less than $0.001$.