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Error Bounds

Guaranteeing approximation within a tolerance · BC Calculus · greeting

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Owl · Approximating with a Taylor polynomial is only half the job. The other half is knowing how wrong you might be. Remainder bounds answer: how many terms guarantee the error stays under 0.001? The Lagrange form of the remainder gives a usable ceiling on |Rₙ(x)| using a bound on the next derivative. That ceiling is a bound — not necessarily the exact error. And the error is not always exactly "the next term," though alternating series sometimes offer that convenience. We'll bound |Rₙ|, choose n for a tolerance, and keep bound vs error distinct.

Owl: Approximating with a Taylor polynomial is only half the job. The other half is knowing how wrong you might be. Remainder bounds answer: how many terms guarantee the error stays under 0.001? The Lagrange form of the remainder gives a usable ceiling on |Rₙ(x)| using a bound on the next derivative. That ceiling is a bound — not necessarily the exact error. And the error is not always exactly "the next term," though alternating series sometimes offer that convenience. We'll bound |Rₙ|, choose n for a tolerance, and keep bound vs error distinct.

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