Re: "Capacitor electronics in ceiling fans create distinct speed "steps" like high, medium, and low. Capacitors control the fan speed in a way that does not produce a hum. Variable fan speed controls, which create a speed control "range" from low to high, are operated by solid state electronics. Variable controls can create a hum because they control the fan's speed in a different manner. For quiet fan operation, capacitor type fan controls are recommended for most ceiling fan installations."
Are there capacitors used in a speed control for a fan, yes there are. Will changing the value of the capacitor allow for different speeds, probably not. It is not just the value of any capacitor that changes the speed of the motor, it's the type of motor used, the topology of the whole circuit, the mechanical load on the motor and the frequency of the voltage applied to the motor.
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"Don't disturb my circles." translation of Archimedes last words
On mine there are 2 caps, the speed controller inserts 1 or the other or both.
As Cuba points out w/o them they hum. I have 3 one with straight wiring. 1 works consistantly, while the other explodes the caps on a regular basis, it now runs with no caps. The big orange store sometimes sells universal replacements.
The exploding cap fan uses approx 9% more power, and gets warm, it also runs slower than the others. Kill-a-Watt says it has a lower P.F., the frequency and voltage are the same.
IIRC the site "all-about circuits" has/had an in depth theory of operation, but the gist is there is a phase shift with voltage and current.