Math 2.0

Approximations and expansions Corollary

Taylor's theorem with the Peano remainder

Builds on Taylor's theoremLittle o notation

Statement

Let k≥1k \ge 1, let II be an open interval, let a∈Ia \in I, and let f ⁣:I→ℝf \colon I \to \mathbb{R} be kk times differentiable at aa. Then f(x)=∑j=0kf(j)(a)j!(x−a)j+o((x−a)k)as x→a.f(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j + o\big((x - a)^k\big) \quad \text{as } x \to a.

Remarks

This is Taylor's theorem written with little o notation. For example, cos⁡x=1−x2/2+o(x3)\cos x = 1 - x^2/2 + o(x^3) as x→0x \to 0. Mathlib states a form for CkC^k functions on a convex set as taylor_isLittleO.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Big O notation, definition, no evidence levelBig O notationdefinitionLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLittle o notation, definition, no evidence levelLittle o notationdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionTaylor's theorem, statement, no evidence levelTaylor's theoremstatementTaylor's theorem with the Peano remainder, statement, no evidence levelTaylor's theorem with…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Big O notation definition
  2. Limit point of a set definition
  3. Little o notation definition
  4. Limit of a function definition
  5. Derivative definition
  6. Differentiable function definition
  7. Derivative function definition
  8. Higher derivative definition
  9. Taylor polynomial definition
  10. Taylor's theorem statement
  11. Taylor's theorem with the Peano remainder statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Statement of the theorem
About this page
Tag03AJHX431A
URIm2://03AJHX431A
Kindstatement, corollary
WritingW0 stub
MSC41A58
StatusThis copy of the site has no build data. Run m2 status 03AJHX431A.
UsesTaylor's theorem, Little o notation

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