Answer
$500$ Terms
Work Step by Step
Recall the Taylor series $\tan^{-1} x= x-\dfrac{x^3}{3}+\dfrac{ x^5}{5}-....; |x| \leq 1$
Now, $| Error|=|\dfrac{(x)^{2n-1}}{2n-1}|$
Set $ x=0.1$; then we have
and $| Error|=|\dfrac{(-1)^n x^n}{n}|=\dfrac{1}{n (10^n)}$
But $\dfrac{1}{2n-1} \lt 10^{-3}$
or, $ \dfrac{1}{2n-1} \lt \dfrac{1}{10^{3}} $
or, $2n \gt 1001$
or, $ n=501$
We need $500$ terms to make sure that the error of magnitude is less than $10^{-3}$.
