What is series connection of capacitors ?

Explanation

Figure shows capacitors `C_(1)` and `C_(2)` combined in series.

The left plate of `C_(1)` and the right plate of `C_(2)` are connected to two terminals of a battery and have charges Q and -Q.
Hence, the right plate of `C_(1)` has charge - Q and the left plate of `C_(2)` induces charge + Q.
Hence, both capacitors have same charge on each capacitor even though they have different capacitance.
Suppose, potential difference between two plates of capacitor `C_(1)` and `C_(2)` are `V_(1)` and `V_(2)` respectively. If potential difference between two ends of combination V, then
`V=_(1)+V_(2)`
`:. V= (Q)/(C_(1))+(Q)/(C_(2)) [ because C=(Q)/(V) implies V = (Q)/(C)]`
`:. (V)/(Q)=(1)/(C_(1))+(1)/(C_(2))`
This combination as an effective capacitor with charge Q and potential difference V. The effective capacitance of the combination is,
`C=(Q)/(V)`
`:.` From equation (2) and (3)
`(1)/(C) =(1)/(C_(1))+(1)/(C_(2))`