How to find value of arctan using Maclaurin

Question

I need to find $\arctan(3)$ and before this, I'm asked to find the Maclaurin series for $\arctan(x)$, so I know they must be related. But $\arctan(x)$ is defined with $x \in [-1,1]$ so what can I do?

Answer 1

Not exactly: $\arctan x$ is defined for all $x$, but the radius of convergence of its Maclaurin series is equal to $1$.

You may reduce the problem to this case, using the relation $$\arctan x+\arctan \frac1x=\begin{cases}\dfrac\pi 2&\text{if }x>0, \\[1ex]-\dfrac\pi 2&\text{if }x<0.\end{cases}$$