Taylor series
Let an agent handle Taylor series by function, centre, terms, convergence and error bounds.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent handle Taylor series by function, centre, terms, convergence and error bounds.
A representation of a function as an infinite sum of terms computed from its derivatives at a single point; centred at zero it is called a Maclaurin series.
What it is for: Approximating functions, analysing behaviour and computing numerically.
It can be compute the series; truncate to a polynomial approximation; bound the remainder; find the radius of convergence.
Distinguishing features
Built from derivatives at a point
Converges within a radius
Remainder terms bound errors
Not all smooth functions equal their series
What it looks like
A sum of terms f^(n)(a)/n! times (x - a)^n.
How it is recognised
Factorials and powers of (x - a)
Maclaurin series when a = 0
Laurent series include negative powers
Related models
is a kind of - category
uses - ingredient
is generalised by - generalisation
is studied in - field
In practice
Families and kinds
Maclaurin series
Taylor polynomials
multivariable Taylor series
Taylor expansions in statistics (delta method)
Failure modes and hazards
Using outside the radius of convergence
Ignoring remainder error
Assuming analyticity
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Series Which expansion.
Function and centre define it.
Function
What is expanded.
Function
Function and centre.
- Which function is expanded, about which point? definition
- What are the first terms? measurement
Terms
Coefficients.
Terms
Coefficients.
- What is the general term? definition
- How are derivatives computed? action
Convergence Where it works.
Convergence limits use.
Radius
Radius of convergence.
Radius
Radius.
- What is the radius of convergence? measurement
- What happens at the boundary? boundary
Analyticity
Equality.
Analyticity
Analyticity.
- Does the series equal the function in that region? boundary
- Is there a counterexample? boundary
Approximation Truncation.
Truncation gives approximations.
Order
Terms kept.
Order
Truncation order.
- How many terms are needed for the required accuracy? measurement
- Over what interval? boundary
Error
Remainder.
Error
Error bound.
- What is the remainder bound? measurement
- Which remainder form is used? definition
Applications Uses.
Taylor series are everywhere.
Numerics
Computation.
Numerics
Numerical use.
- How is the series used in numerical methods? definition
- Is it stable? boundary
Physics
Linearisation.
Physics
Linearisation.
- How does linearisation use the first terms? definition
- When is it valid? boundary
What the second pass must settle
- Should Taylor polynomials be a separate entry?
- How should standard expansions be listed?
- How should the statistics use be linked?