Math 2.0

Approximations and expansions Theorem

Taylor's theorem with the integral remainder

Builds on Taylor polynomialFundamental theorem of calculusIntegration by parts

Statement

Let k≥0k \ge 0, let a≠xa \ne x be real numbers, and let JJ be the closed interval with endpoints aa and xx. Let ff be a real function on JJ such that f(k+1)f^{(k+1)} exists and is continuous on JJ. Then f(x)=Pk(x)+∫axf(k+1)(t)k!(x−t)k dt,f(x) = P_k(x) + \int_a^x \frac{f^{(k+1)}(t)}{k!} (x - t)^k \, dt, where PkP_k is the Taylor polynomial of degree kk of ff at aa.

Remarks

For k=0k = 0 it is the fundamental theorem of calculus, f(x)=f(a)+∫axf′(t) dtf(x) = f(a) + \int_a^x f'(t)\, dt. Repeated integration by parts gives the general case. The hypotheses can be weakened to absolute continuity of f(k)f^{(k)}. Mathlib states it as taylor_integral_remainder_of_absolutelyContinuous.

Reading path

To understand this page from the ground up, read these 45 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Antiderivative definition
  19. Differentiable function definition
  20. Fermat's theorem on stationary points statement
  21. Riemann integral definition
  22. Convergent sequence definition
  23. Monotone convergence theorem for sequences statement
  24. Every sequence has a monotone subsequence statement
  25. Intermediate value theorem statement
  26. Differentiable functions are continuous statement
  27. Derivative function definition
  28. Definite integral definition
  29. Bolzano–Weierstrass theorem statement
  30. Product rule statement
  31. Boundedness theorem statement
  32. Higher derivative definition
  33. Additivity of the integral over intervals statement
  34. Monotonicity of the integral statement
  35. Extreme value theorem statement
  36. Continuously differentiable function definition
  37. Taylor polynomial definition
  38. Mean value theorem for integrals statement
  39. Rolle's theorem statement
  40. Mean value theorem statement
  41. First fundamental theorem of calculus statement
  42. Second fundamental theorem of calculus statement
  43. Integration by parts statement
  44. Fundamental theorem of calculus statement
  45. Taylor's theorem with the integral remainder statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Explicit formulas for the remainder
About this page
Tag0WJGR81HBD
URIm2://0WJGR81HBD
Kindstatement, theorem
WritingW0 stub
MSC41A80
StatusThis copy of the site has no build data. Run m2 status 0WJGR81HBD.
UsesTaylor polynomial, Fundamental theorem of calculus, Integration by parts

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