Firstly: your 8200 µF capacitor would be an electrolytic capacitor. These are usually tolerated -20% ... + 50 % Also they have a considerable series resistance (ESR). Completely (my opinion) unsuitable for a reasonable LC filter.
But let's assume you had a perfect 8200 µF capacitor...
What size variable capacitor should I use in parallel with the cap.?
That would be one big capacitor...
For 5 Hz you'd need ~20000 µF.
For 50 Hz you'd need ~200 µF.
Since you want to use the variable capacitor in parallel to the fixed one, you'd use a fixed 200 µF capacitor and you'd require a 0 µF ... 20000 µF variable capacitor. You will not find such a beast.
Your 8200 µF capacitor will be useless above the 7.8 Hz you calculated as any other capacitor in parallel will increase the total capacitance and thus lower the frequency. Increasing the frequency is noz possible by paralleling capacitors.
For creating a reasonably accurate big capacitance, you can use a known good small capacitor plus an opamp to form a
capacitance multiplier.
Another approach would be to use a fixed, known good capacitor of a small value plus a much larger inductor. Of course, a bigger inductor will be unwieldy. But rescue comes agian in the form of opamps. You can
simulate a large inductance using capacitors, resistors and opamps.
With these circuits you can adjust the frequency by adjusting resistors (e.g. by using a potentometer).
