How would you prove the formula for the total capacitance of a system consisting of several capacitors linked in series?

The first thing you need to do is to decide what is kept constant within the system. (P.S. Essentially all problems in physics require conservation laws to some extent)
In the case of components in series, you know that the current I flowing through the capacitors should be mainained constant. But this is the same with saying that on each capacitor there will be storred the same amount of charge, q. It follows that the total voltage drop across the system is U = q / C = q / C1 + q / C2 + Q / C3 + ...