Capacitors in Series

In a series combination, capacitors $C_1, C_2, C_3, \ldots$ are connected end to end in a single chain, with only the two OUTER ends of the chain connected to the source of potential difference $V$.

Same charge on every capacitor. Consider, for simplicity, two capacitors $C_1$ and $C_2$ in series. The inner plate connecting them (the right plate of $C_1$ joined to the left plate of $C_2$) is electrically isolated from the rest of the circuit -- charge cannot flow onto or off this isolated inner conductor from anywhere else, so it must carry exactly zero NET charge; but if the outer plate of $C_1$ carries charge $+Q$, it must induce an equal $-Q$ on the near face of that isolated inner conductor, which in turn forces an equal and opposite $+Q$ to appear on its far face (facing $C_2$), and so on down the chain. The upshot is that every capacitor in a series chain carries exactly the same charge magnitude $Q$ -- the same charge that flows from the battery's terminals into the two outermost plates.

Voltages add. The total potential difference supplied by the source is shared, unequally in general, across the individual capacitors:

using $V_i = Q/C_i$ for each capacitor, since $Q$ is common to all of them. Defining the equivalent capacitance $C_s$ by $V = Q/C_s$, and dividing the equation above through by $Q$,

$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots$ …