series
Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.
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.
Researched by: Claude
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Terms What is summed.
Term rule
General term.
Term rule
The general term.
- What is the general term of the series? definition
- Where does the summation start? definition
Kind
Geometric, power, Fourier.
Kind
The kind of series.
- What kind of series is this? definition
- Is it formal or analytic? boundary
Convergence Does it sum.
Tests
Ratio, root, comparison.
Convergence test
Test used.
- Does the series converge, by which test? measurement
- For which values of the variable? boundary
Type of convergence
Absolute, conditional, uniform.
Convergence type
The type of convergence.
- Is the convergence absolute, conditional or uniform? definition
- What operations does that permit? boundary
Sum The value.
Value
Closed form.
Sum
The sum.
- What is the sum, in closed form if possible? measurement
- How was it derived? provenance
Approximation
Truncation error.
Error bound
Error of partial sums.
- How many terms give the needed accuracy? measurement
- What bounds the truncation error? measurement
Application Where it is used.
Expansion
Functions as series.
Expansion
Function expansions.
- Which function does this series represent, and where? definition
- What is the radius of convergence? measurement
Computation
Numerical use.
Numerics
Numerical behaviour.
- Is the series numerically stable to sum? boundary
- Should acceleration methods be used? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.REL
- Other names and narrower kinds
- formal power series, ultrapolynomial, quasisymmetric function, asymptotic expansion, Cornish–Fisher expansion, operator product expansion, convergent series, multipole expansion, basic hypergeometric series, Charlier series, geometric series, function series
What it is Filled
In mathematics, the sum of the terms of a sequence, written a1 + a2 + a3 + ..., which may be finite or infinite and, if infinite, convergent or divergent.
Why it exists Filled
Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.
Distinguishing features Filled
- A sum of terms, not the sequence itself
- Infinite series may converge or diverge
- Formal power series need not converge
- Asymptotic expansions can be useful even when divergent
What robots and AI may and may not do Filled
Must not
- Sum a divergent series as if it were convergent without stating the method.
- Rearrange a conditionally convergent series and present the result as its sum.
- Truncate a series without stating the error.
Only with a human decision
- Relying on a truncated series in safety-critical calculations.
May
- Determine and state whether a series converges, with the test used.
- Truncate a series only with an error bound.
Moral aspects Filled
- Mathematical errors in engineering and finance hurt people who trust the result.
Who is affected
- Users of results
- Students
Owners Filled
Steward
Nobody owns the concept; it is held in common.
Links to other meta-models Filled
parent
- Q177239 - registry parent class
- Q218005 - registry parent class
- Q6029858 - registry parent class
related
- sequence - terms of a sequence
- analysis and physics - approximation and expansion
- series as a set of works - a homonym
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.series
- Wikidata: Q170198 (https://www.wikidata.org/wiki/Q170198)
Direct properties not applicable Not applicable
Not applicable
Plane XCT: no invented physical properties.
Recognition optional Filled
- Sigma notation
- Named series such as geometric, harmonic, Taylor, Fourier
- TV and book series are different senses
- Summation notation with a sigma, or terms with plus signs and ellipsis.
Capabilities and actions required Filled
- sum it, when it converges
- test convergence, with standard tests
- expand functions as series, such as power or Fourier series
- truncate it, for approximation, with an error bound
Hazards and failure modes required Filled
- Summing a divergent series as if convergent
- Rearranging conditionally convergent series
- Truncating without an error bound
Standards and interfaces required Filled
- ISO 80000-2 notation
Context of use required Filled
- Representing functions, approximating values and solving problems in analysis, physics and engineering.
- geometric and arithmetic series
- power, Taylor and Laurent series
- Fourier series
- formal power series
- asymptotic expansions
Sources Filled
- Wikidata item Q170198: series - identity and sense of the item
- Wikipedia: Series (mathematics) - general description of the item
Open questions
- Should named series be separate entries?
- How should convergence conditions be recorded?
- How should formal and analytic series be distinguished?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q170198/spec.json