Math 2.0

definition

Taylor series

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.

Explanations

L1 human

The Taylor series is the Taylor polynomial of infinite degree. It uses all the derivatives of ff at one point.

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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
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  14. Continuously differentiable function definition
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  16. Taylor series definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Taylor_series, Definition

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