Answer
$500$ Terms
Work Step by Step
The Taylor series for $\tan^{-1} x $ can be defined as: $\tan^{-1} x= x-\dfrac{x^3}{3}+\dfrac{ x^5}{5}-....; |x| \leq 1$
We have $| Error|=|\dfrac{(x)^{2n-1}}{2n-1}|$
When $ x=0.1$, we have:
$| Error|=|\dfrac{(-1)^n x^n}{n}|=\dfrac{1}{n (10^n)}$
But $\dfrac{1}{2n-1} \lt 10^{-3} \implies \dfrac{1}{2n-1} \lt \dfrac{1}{10^{3}} $
or, $2n \gt 1001 \implies n=501$
So, we have to consider $500$ terms to make sure that the error of magnitude is less than $10^{-3}$.
