Answer
$$n \geq 31$$
Work Step by Step
Consider: $S_n=a_1-a_2+......+(-1)^{n+1}a_n$
Now, $$ |S-S_n| \leq |a_{n+1}| \\ \implies |\space Error | \lt 0.001 =|\dfrac{1}{(n+1)^2+3}| \lt 0.001 \\ \implies (n+1)^2+3 \gt 1000 \\ \implies n \geq 31$$
Thomas' Calculus 13th Edition
by Thomas Jr., George B.
Published by
Pearson
ISBN 10:
0-32187-896-5
ISBN 13:
978-0-32187-896-0
Chapter 10: Infinite Sequences and Series - Section 10.6 - Alternating Series and Conditional Convergence - Exercises 10.6 - Page 603: 53
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