A weird sum

Relevant wiki: Maclaurin Series

$\begin{aligned} S & = 4 - \frac 43 + \frac 45 - \frac 47 + \cdots \\ & = 4 \color{#3D99F6} \left(1- \frac 13 + \frac 15 - \frac 17 + \cdots \right) & \small \color{#3D99F6} \text{By Maclaurin series} \\ & = 4\ {\color{#3D99F6} \tan^{-1} 1} = 4 \times \frac \pi 4 = \pi \\ & \approx \boxed{3.142} \end{aligned}$

The function can be rewritten as: $4\sum_{n=0}^{∞}\frac{(-1)^{n+1}}{(2n-1)}$ .
This is the Taylor Series for $4arctan(x)$ .
Thus, the answer is $4arctan(∞)-4arctan(1)=4\frac{\pi}{2}-4\frac{\pi}{4}=4\frac{\pi}{4}=\pi$ .