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Alternating Series

Sign flips, conditional convergence, and remainders · BC Calculus · greeting

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Owl · When signs flip + − + −, cancellation can rescue a series that would otherwise diverge. Alternating series ask: can those cancellations save a barely dying positive skeleton? Absolute convergence — Σ |aₙ| converges — is the strong form. Conditional convergence means the signed series converges but the absolute version does not. They are not the same idea wearing different clothes. We'll run the alternating series test checklist, estimate remainders, and build examples that separate absolute from conditional. Two traps to dodge: "alternating always converges," and "conditional equals absolute."

Larry’s gold answer — Show text to read it out loud.

Larry · gold answer · For Σ (−1)^{n+1}/n, the magnitudes 1/n decrease to 0, so AST should say it converges — even though the harmonic series diverges.

Owl: When signs flip + − + −, cancellation can rescue a series that would otherwise diverge. Alternating series ask: can those cancellations save a barely dying positive skeleton? Absolute convergence — Σ |aₙ| converges — is the strong form. Conditional convergence means the signed series converges but the absolute version does not. They are not the same idea wearing different clothes. We'll run the alternating series test checklist, estimate remainders, and build examples that separate absolute from conditional. Two traps to dodge: "alternating always converges," and "conditional equals absolute."

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