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In a series RLC circuit, resonance occurs at ω₀ = 1/√(LC) where inductive and capacitive reactances cancel, leaving only resistance. At resonance, impedance is minimum (Z = R), current is maximum, voltage across the coil and capacitor are equal in magnitude but 180° out of phase, and voltage and current are in phase. Series resonance is exploited in bandpass filters, tuned amplifiers, and impedance matching.
From your study of impedance and admittance, you know that inductors and capacitors both oppose current flow, but in opposite ways that depend on frequency. An inductor's reactance X_L = ωL grows with frequency; a capacitor's reactance X_C = 1/(ωC) shrinks with frequency. Connect them in series and you have two frequency-dependent opponents. At one special frequency they cancel exactly — that is resonance, and the circuit's behavior at that frequency is dramatically different from any other.
At the resonant frequency ω₀ = 1/√(LC), the total reactance is X_L − X_C = ω₀L − 1/(ω₀C) = 0. The series impedance reduces to Z = R — purely resistive, as if the inductor and capacitor weren't there. Since Z is at its minimum, the current amplitude I = V_s/R is at its maximum. All of the source voltage appears across the resistor; none is "wasted" fighting reactive elements. This maximum-current condition is why resonance is so useful: you can extract maximum power transfer from a source at one specific tunable frequency.
The voltages across the inductor and capacitor at resonance are not zero — they can actually be *much larger* than the source voltage. At ω₀, V_L = I·X_L = (V_s/R)·ω₀L, which exceeds V_s whenever ω₀L > R. This voltage amplification factor is the quality factor Q = ω₀L/R = 1/(ω₀CR). A high-Q circuit (large L/R or small R) has sharp resonance: voltage across L and C can be many times the input voltage, and the circuit responds strongly only to a narrow band of frequencies near ω₀. A low-Q circuit has broad, weak resonance. The voltage across C and L are equal in magnitude at ω₀ but exactly 180° out of phase, so they cancel in series while each independently reaches Q times the source voltage.
This behavior defines the bandpass character of a series RLC filter. Near ω₀, low impedance allows large current and large output across R. Far from ω₀ — either very low frequencies where the capacitor dominates and blocks current, or very high frequencies where the inductor dominates and chokes current — the impedance rises and the current falls. The bandwidth of the passband is BW = R/L = ω₀/Q: narrow for high-Q circuits, wide for low-Q. Practical applications include radio tuners (selecting one station's frequency while rejecting others), antenna impedance matching, and intermediate-frequency (IF) amplifier stages in radio receivers — all of which exploit the frequency selectivity that resonance provides.