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Ordered lists and whether they settle · BC Calculus · greeting

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Owl · A sequence is just an ordered list of numbers — a₁, a₂, a₃, … — indexed by the positive integers. Think of successive snapshots of a fading echo, or daily readings on a sensor as it cools toward room temperature. Convergence means those snapshots approach a single number L. Divergence means they don't: they wander, oscillate, or escape to infinity. Before any algebra, picture aₙ = 1/n. Where is that heading? And is "bounded" the same as "converges"? We'll plot terms, practice limit tricks, and meet the monotone-and-bounded guarantee — plus a trap about recursive updates that never settle.

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Larry · gold answer · For 1/n, the terms get smaller and seem to head toward 0. So lim aₙ = 0 — the sequence converges to 0.

Owl: A sequence is just an ordered list of numbers — a₁, a₂, a₃, … — indexed by the positive integers. Think of successive snapshots of a fading echo, or daily readings on a sensor as it cools toward room temperature. Convergence means those snapshots approach a single number L. Divergence means they don't: they wander, oscillate, or escape to infinity. Before any algebra, picture aₙ = 1/n. Where is that heading? And is "bounded" the same as "converges"? We'll plot terms, practice limit tricks, and meet the monotone-and-bounded guarantee — plus a trap about recursive updates that never settle.

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