Chapter 10: Infinite Sequences and Series - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 634: 57

4 terms

Recall the Taylor series for $\tan^{-1} x= x-\dfrac{x^3}{3}+\dfrac{ x^5}{5}-....; |x| \leq 1$ Now, $| Error|=(48) |\dfrac{(-1)^{n-1} (\dfrac{1}{18})^{2n-1}}{2n-1}| \lt \dfrac{1}{3}(10^{-6})$ and $48|\dfrac{ (\dfrac{1}{18})^{2n-1}}{2n-1}| \lt \dfrac{1}{3}(10^{-6})$ $\implies n \geq 4$ Thus, the required number of terms are: $4$.