##### Question 15.4.3

Using the fact that $\frac{\dif }{\dif x}\tan^{-1}x = \frac{1}{1+x^2}$ which is the sum of the infinite geometric series $1 + (-x^2) + (-x^2)^2 + (-x^2)^3 + \cdots$ show that $\tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$ This series was discovered by the Indian mathematician-astronomer Madhava of Sangamagrama, and used to compute $\pi$ to astonishing precision (see Question 11.4.17). Show answer