What does it mean for a capcitor to short-circuit in a circuit that consists of an inductor, a capacitor, and a voltage source (no resistor)?

 

What you want to do is analyze the effective impedance loading the voltage source.

The impedance of a capacitor is Z(jw) = 1/jwC
and the impedance of an inductor is Z(jw) = jwL

j = (-1)^0.5 and w = frequency in radians/sec

The inductor and capacitor will either be in series or parallel and using the above equations you can derive an equation for the total impedance loading the voltage source:

Notice at high frequencies the impedance of a capacitor is 0 (it's a short), and at high frequencies the impedance of a capacitor is infinity (it's open).

The opposite is true for an inductor.

Usually an inductor, capacitor and resistor are used in parallel to realize a tank circuit: At the resonant frequency, w = (L*C)^-0.5, the impedance of the inductor and the impedance of the capacitor exactly cancel eachother out leaving only the impedance of the resistor. So what you can make is a filter circuit by appropriatley picking L and C values to allow a certain frequency signal to pass and all other frequencies to be blocked.

I believe the author of this website has a section on tank circuits, yuo can read that for more information.