In an LC circuit, the resonance frequency is given by
[tex]\omega_0 = \frac{1}{ \sqrt{LC} } [/tex]
where L is the value of the inductance and C is the value of the capacitance.
If we double the capacitance: C'=2C and the inductance is halved: L'=L/2, the new resonance frequency is
[tex]\omega' = \frac{1}{ \sqrt{L'C'} }= \frac{1}{ \sqrt{ \frac{L}{2} \cdot 2C } }= \frac{1}{ \sqrt{LC} }=\omega_0 [/tex]
so, the new frequency of resonance is equal to the original value, so it didn't change.
