Chapter 9 - Section 9.10 - The Binomial Series and Applications of Taylor Series - Exercises - Page 550: 57

$4$ Terms

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 $| Error|=48|\dfrac{(-1)^{n-1} (\dfrac{1}{18})^{2n-1}}{2n-1}| \lt \dfrac{1}{3}(10^{-6})$ Now, $48|\dfrac{ (\dfrac{1}{18})^{2n-1}}{2n-1}| \lt \dfrac{1}{3}(10^{-6})$ or, $ n \geq 4$ So, the number of terms is $4$.