Math 2.0

Approximations and expansions Definition

Taylor series

Builds on Smooth functionPower seriesTaylor polynomial

Definition

Let II be an open interval, let a∈Ia \in I, and let f ⁣:I→ℝf \colon I \to \mathbb{R} be smooth. The Taylor series of ff at aa is the power series ∑n=0∞f(n)(a)n!(x−a)n.\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n.

Remarks

Its partial sums are the Taylor polynomials of ff at aa. The Taylor series may diverge for each x≠ax \ne a. It may also converge to a function other than ff, as for the smooth non-analytic function. The Taylor series converges to ff near aa exactly when ff is analytic at aa. For a=0a = 0 it is the Maclaurin series.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionSeries, definition, no evidence levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPower series, definition, no evidence levelPower seriesdefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionTaylor series, definition, no evidence levelTaylor seriesdefinition
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Derivative definition
  7. Series definition
  8. Continuous function definition
  9. Differentiable function definition
  10. Power series definition
  11. Derivative function definition
  12. Higher derivative definition
  13. Taylor polynomial definition
  14. Continuously differentiable function definition
  15. Smooth function definition
  16. Taylor series definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor_series, Definition
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Kinddefinition
WritingW1 stated
MSC41A58
DefinesTaylor series
UsesSmooth function, Power series, Taylor polynomial

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