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10.12 Lagrange Error Bound

Part of Infinite Sequences and Series in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The Lagrange error bound gives a maximum possible size for the difference between a function and its Taylor polynomial approximation. If Tn(x){T_n(x)} is the nth Taylor polynomial for f(x) centered at a, then the remainder satisfies
∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}
where M is an upper bound on ∣f(n+1)(z)∣{|f^{(n+1)}(z)|} for values z between a and x. This tells you how far off the Taylor approximation could be at a given point.

You use this bound by finding the next derivative, choosing a reasonable maximum M, and plugging into the formula. The result is an overestimate, not usually the exact error, but it guarantees that the true error is no larger than that amount. This is useful when you want to justify how accurate a Taylor polynomial is or how many terms are needed to achieve a desired accuracy. The Lagrange error bound is especially important because it turns Taylor approximation into a rigorous approximation method rather than just a guess.

In this unit

15 lessons

  1. 10.1Defining Convergent and Divergent Infinite SeriesPreview lesson
  2. 10.2Working with Geometric SeriesPreview lesson
  3. 10.3The nth Term Test for DivergencePreview lesson
  4. 10.4Integral Test for ConvergencePreview lesson
  5. 10.5Harmonic Series and p-SeriesPreview lesson
  6. 10.6Comparison Tests for ConvergencePreview lesson
  7. 10.7Alternating Series Test for ConvergencePreview lesson
  8. 10.8Ratio Test for ConvergencePreview lesson
  9. 10.9Determining Absolute or Conditional ConvergencePreview lesson
  10. 10.10Alternating Series Error BoundPreview lesson
  11. 10.11Finding Taylor Polynomial Approximations of FunctionsPreview lesson
  12. 10.12Lagrange Error BoundThis lesson
  13. 10.13Radius and Interval of Convergence of Power SeriesPreview lesson
  14. 10.14Finding Taylor or Maclaurin Series for a FunctionPreview lesson
  15. 10.15Representing Functions as Power SeriesPreview lesson

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