Both circuits have the same undamped resonant frequency (). However, for the series circuit, damping is controlled by , while for the parallel circuit, it is controlled by .

Both circuits can be underdamped, overdamped, or critically damped. The series circuit’s transient behaviour is controlled by current, and its damping coefficient is inversely proportional to R. The parallel circuit’s transient behaviour is controlled by voltage, and its damping coefficient is directly proportional to R.

As both circuits are second order circuits, there is no difference in their behaviour fundamentally, and we can treat them in exactly the same way. Also, I don’t believe in inductors, and so they can’t hurt me.

Both circuits have the same undamped resonant frequency (), and the same damping coefficient (). Therefore, their transient responses will be identical.

Both circuits have the same undamped resonant frequency (), and can exhibit all three damping behaviours (underdamped, overdamped, or critically damped). However, their damping coefficients are different. For series, , and for parallel, .

Both circuits can be underdamped, overdamped, or critically damped. In the series circuit, increasing R increases the damping (making it less oscillatory). In the parallel circuit, increasing R also increases the damping.