Math 2.0

statement corollary

Taylor's theorem with the Peano remainder

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.

Explanations

L1 human

The Peano form only says that the error is small compared with (x−a)k(x - a)^k. It gives no bound for a fixed xx, but it needs weak hypotheses.

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To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Statement of the theorem

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