Math 2.0

Approximations and expansions Definition

Maclaurin series

Builds on Taylor series

Definition

Let II be an open interval with 0∈I0 \in I, and let f ⁣:I→ℝf \colon I \to \mathbb{R} be smooth. The Maclaurin series of ff is its Taylor series at 00: ∑n=0∞f(n)(0)n!xn.\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n.

Remarks

For example, the Maclaurin series of exe^x is ∑xn/n!\sum x^n / n!, and the one of 1/(1−x)1/(1 - x) is ∑xn\sum x^n. The series is named after Colin Maclaurin.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
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 seriesdefinitionMaclaurin series, definition, no evidence levelMaclaurin seriesdefinition
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. 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
  17. Maclaurin series definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor_series, Definition
About this page
Tag043SJ5RF59
URIm2://043SJ5RF59
Kinddefinition
WritingW1 stated
MSC41A58
DefinesMaclaurin series
UsesTaylor series

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by