How is positive feedback coupled to the input in a Colpitts oscillator?

A Colpitts oscillator relies on an LC tuned circuit to determine its operating frequency, much like the Hartley oscillator. However, the mechanism it uses to route a portion of the signal back to the input to sustain oscillation is distinct. The Colpitts oscillator achieves this positive feedback by using a capacitive voltage divider. This divider consists of two capacitors placed in series across the inductor. The center connection (tap) between these two capacitors is used to feed the output signal back to the input with the correct phase. Distractors like a 'tapped coil' apply to the Hartley design. A handy memory trick is to associate the 'C' in Colpitts with the 'C' in Capacitors.