Core of the low mass stars never get hot enough for carbon fusion thus can the fusion of carbon occur in intermediate- and high-mass stars but not in low-mass stars.
Since the low mass stars cannot get hot enough for fusion, they have long lives and thus live as white dwarfs. The stars become more luminous over time since it consumes it core hydrogen and converts it to helium. The core shrinks and heats up gradually. After the helium has fused into carbon, the core collapses again and the outer layer of the stars are expelled.
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