Math 2.0

statement theorem

Taylor's theorem with the Cauchy remainder

Statement

Let k≥0k \ge 0 and let a≠xa \ne x be real numbers. Let JJ be the closed interval with endpoints aa and xx. Let ff be a real function with f(k)f^{(k)} continuous on JJ and f(k+1)f^{(k+1)} existing on the interior of JJ. Then there is ξ\xi strictly between aa and xx with f(x)=Pk(x)+f(k+1)(ξ)k!(x−ξ)k(x−a),f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{k!} (x - \xi)^k (x - a), where PkP_k is the Taylor polynomial of degree kk of ff at aa.

Remarks

Apply the mean value theorem to F(t)=f(t)+∑j=1kf(j)(t)(x−t)j/j!F(t) = f(t) + \sum_{j=1}^{k} f^{(j)}(t) (x - t)^j / j! on JJ. Note that F(x)=f(x)F(x) = f(x) and F(a)=Pk(x)F(a) = P_k(x). This form sometimes gives better bounds than the Lagrange form, for example for ln⁡(1+x)\ln(1 + x) with −1<x<0-1 < x < 0. Mathlib states it as taylor_mean_remainder_cauchy.

Explanations

L1 human

The Cauchy form is another exact expression for the error. It moves part of the power of (x−a)(x - a) to a power of (x−ξ)(x - \xi).

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementHigher derivative, definition, no evidence levelHigher derivativedefinitionBoundedness theorem, statement, no evidence levelBoundedness theoremstatementTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementTaylor's theorem with the Cauchy remainder, statement, no evidence levelTaylor's theorem with…statement
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  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Limit of a sequence definition
  12. Monotone sequence definition
  13. Continuous function definition
  14. Differentiable function definition
  15. Fermat's theorem on stationary points statement
  16. Convergent sequence definition
  17. Monotone convergence theorem for sequences statement
  18. Every sequence has a monotone subsequence statement
  19. Derivative function definition
  20. Bolzano–Weierstrass theorem statement
  21. Higher derivative definition
  22. Boundedness theorem statement
  23. Taylor polynomial definition
  24. Extreme value theorem statement
  25. Rolle's theorem statement
  26. Mean value theorem statement
  27. Taylor's theorem with the Cauchy remainder statement

Sources

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Explicit formulas for the remainder

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