Question

expression for resonance frequency of series LCR circuit

Answer 1

current in a series LCR circuit = I = V / Z

$Z=\sqrt{R^2+(\omega L-\frac{1}{\omega C})^2}$

So Z = impedance is minimum when  ω L = 1 / (ω C) 
   ie.,  ω = 1 /√ (L C)

When impedance is minimum, then the current is maximum and that situation is called resonance of the circuit.

So resonance happens, when the frequency of the AC voltage source in the series LCR circuit is equal to 1/√(LC).