A resistor-inductor-capacitor combination , , , has a resonant frequency that is just the same as that of a different combination , , . The two are next connected in combination in series. We have to show that this new circuit also has the same resonant frequency as the separate individual circuits.

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The resonant frequencies of the two LCR circuits with circuit elements , , and , , are same. Therefore,

.

When the two LCR circuits are combined in series the inductance of the combination will be

,

and the capacitance of the combination will be

.

Therefore, the resonant frequency of the series combination of the two LCR circuits will be

Therefore,

The resonant frequency of the series combination of the two LCR circuits each of resonant frequency is also .