Math 2.0

Approximations and expansions Theorem

Taylor's theorem

Builds on Taylor polynomial

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. Let PkP_k be the Taylor polynomial of degree kk of ff at aa. Then there is a function hk ⁣:I→ℝh_k \colon I \to \mathbb{R} with hk(x)→0h_k(x) \to 0 as x→ax \to a and f(x)=Pk(x)+hk(x)(x−a)kfor all x∈I.f(x) = P_k(x) + h_k(x) (x - a)^k \quad \text{for all } x \in I.

Remarks

In little o notation, the conclusion is the Peano form f(x)−Pk(x)=o((x−a)k)f(x) - P_k(x) = o((x - a)^k). Under stronger hypotheses, the remainder has explicit forms: the Lagrange form, the Cauchy form, and the integral form. For k=1k = 1, it says that f(x)=f(a)+f′(a)(x−a)+o(x−a)f(x) = f(a) + f'(a)(x - a) + o(x - a).

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit 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 theoremstatement
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Derivative definition
  4. Differentiable function definition
  5. Derivative function definition
  6. Higher derivative definition
  7. Taylor polynomial definition
  8. Taylor's theorem statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor%27s_theorem, Statement of the theorem
About this page
Tag03QHBVW4TT
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Kindstatement, theorem
WritingW1 stated
MSC41A58
StatusThis copy of the site has no build data. Run m2 status 03QHBVW4TT.
UsesTaylor polynomial
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