power series
Let an agent handle power series by coefficients, centre, radius of convergence and the function represented.
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 power series by coefficients, centre, radius of convergence and the function represented.
An infinite series of the form sum of a_n (x - c)^n, representing a function near a centre c within a radius of convergence; Taylor series and hypergeometric series are examples.
What it is for: Representing and approximating functions in analysis, physics and computation.
It can be compute coefficients; find the radius of convergence; differentiate and integrate term by term within it; truncate for approximation, with error bounds.
Distinguishing features
Powers of (x - c) with coefficients
Converges inside a disc of radius R
Term-by-term operations valid inside
Formal and analytic uses differ
What it looks like
Sigma notation with powers of (x - c).
How it is recognised
Terms with increasing powers
Named series such as hypergeometric or Lidstone series
Formal power series need not converge
Related models
is a kind of - function series
represents - use
is studied by - analysis
is confused with - finite
In practice
Families and kinds
Taylor and Maclaurin series
hypergeometric series
Lidstone series
formal power series
Standards and regulation
ISO 80000-2
Failure modes and hazards
Using the series outside its radius
Slow convergence near the boundary
Round-off in numerical summation
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.
Definition Coefficients and centre.
Coefficients and centre define the series.
Coefficients
Rule for a_n.
Coefficients
Coefficient rule.
- What is the rule for the coefficients? definition
- What is the centre? definition
Kind
Taylor, hypergeometric.
Kind
Kind of series.
- Which named series is this? definition
- Is it formal or analytic? boundary
Convergence Where it works.
Radius of convergence is essential.
Radius
R.
Radius
Radius of convergence.
- What is the radius of convergence? measurement
- What happens on the boundary? boundary
Behaviour
Speed of convergence.
Behaviour
Convergence speed.
- How fast does it converge at a given point? measurement
- How many terms are needed? measurement
Function What it represents.
Series represent functions.
Represented function
Sum.
Function
The function.
- Which function does the series sum to? definition
- Is it analytic there? boundary
Operations
Term-by-term.
Operations
Valid operations.
- Can it be differentiated or integrated term by term here? boundary
- What results? definition
Computation Numerical use.
Numerics need care.
Truncation
Error.
Error bound
Truncation error.
- What is the truncation error? measurement
- Which bound applies? definition
Stability
Round-off.
Stability
Numerical stability.
- Is summation numerically stable? boundary
- Should another method be used? action
What the second pass must settle
- Should named power series be separate entries?
- How should convergence data be recorded?
- How should formal series be marked?