Unit 15: How far can you trust a Taylor polynomial?
About this unit
A Taylor polynomial's error is roughly its first missing term, at most M|x − a|ⁿ⁺¹/(n + 1)!, and some series only converge within a fixed distance of their centre.
By the end, you can: Measure how a Taylor polynomial's error grows with distance, bound it using the next derivative, and see why some series only work on an interval.
Modules
5 modulesLearn
- The error is the first term you left outArticle · Near the centre the error of Pₙ behaves like the (n + 1)th term, so it scales like (x − a)ⁿ⁺¹, and the remainder bound M|x − a|ⁿ⁺¹/(n + 1)! makes that safe.Not started
- Series that give upArticle · The series for 1/(1 − x) and ln(1 + x) only converge for |x| < 1 (ln also at x = 1), because the terms grow beyond that; eˣ and sin x converge everywhere.Not started
- Push ln(1 + x) to its limitExperiment · Raise the degree of the ln(1 + x) polynomial and watch the good band widen on the left but stall just short of x = 1 on the right.Not started
- Trusting a Taylor polynomialReview · Error ≈ the first missing term, bounded by M|x − a|ⁿ⁺¹/(n + 1)!; a series works only inside its radius of convergence.Not started