We select the left side AC circuit.
We write the potential difference across each network element and sum them.
Z = impedance in the circuit is minimum when
Resonance occurs at this frequency ω₀. Then the impedance Z = R and the output voltage
For other values of ω, Z increases and so v_out decreases. Hence the circuit acts as a band pass filter.
The band width can be obtained by finding the ω₁ and ω₂, when the power is 1/2 the maximum and when v_out = v_m /√2 .
Then it works as a good band pass filter ,