Nested Circles

If we continue drawing circles, we are going to obtain at the $n^{th}$ step $n!$ circles with radius $\large\frac{1}{n!}$ each. Thus, the total area of $\large n!$ circles is : $\large n! \times \frac{\pi}{(n!)^2}=\frac{\pi}{n!}$

The desired area is obtained by adding the areas in step $2$ , then subtracting those at step $3$ , then adding the areas at step $4$ and then subtracting those at step $5$ and so forth.

To solve it : $\large\frac{\pi}{2!} - \frac{\pi}{3!} + \frac{\pi}{4!} - ...= \pi ( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}) + ...= \pi\times e^{-1} = \frac{\pi}{e}$