Step 1: Recall the resonance frequency formula for an LCR circuit.
In an LCR series AC circuit, resonance occurs when the inductive reactance equals the capacitive reactance. The resonance frequency is: \[ f_0 = \frac{1}{2\pi\sqrt{LC}} \] At resonance, impedance is minimum and current is maximum.
Step 2: Identify the new values of L and C.
The inductance is reduced to $\frac{1}{4}$ of its original value: $L' = \frac{L}{4}$. The capacitance is increased to 16 times: $C' = 16C$.
Step 3: Write the new resonance frequency.
\[ f' = \frac{1}{2\pi\sqrt{L'C'}} = \frac{1}{2\pi\sqrt{\frac{L}{4} \times 16C}} \]
Step 4: Simplify the expression under the square root.
\[ \frac{L}{4} \times 16C = 4LC \] Therefore: \[ f' = \frac{1}{2\pi\sqrt{4LC}} = \frac{1}{2\pi \times 2\sqrt{LC}} = \frac{1}{2} \times \frac{1}{2\pi\sqrt{LC}} \]
Step 5: Express in terms of $f_0$.
Since $f_0 = \frac{1}{2\pi\sqrt{LC}}$: \[ f' = \frac{f_0}{2} \] The product $LC$ increased by a factor of 4, so $\sqrt{LC}$ increased by a factor of 2, and the frequency decreased by a factor of 2.
Step 6: State the final answer.
The new resonance frequency is: \[ \boxed{\frac{f_0}{2}} \]