Expand $\tan^{-1}x$ as infinite series

Question

How can we obtain the infinite series for $\tan^{-1}(x)?$
Finding the derivatives in Taylor series becomes difficult.

Answer 1

You can use Taylor, Maclaurin series to expand it, but I find the following very easy.
$$f(x)=tan^{-1}x$$ $$f'(x)=\frac{1}{1+x^2}=1-x^2+x^4-x^6+...$$
Integrating both sides,
$$f(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+...$$