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Research draft

series

vr.tr.series · XCT.REL

Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.

Thing Registry Cross-cutting context

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 series with their terms, convergence and sums, and separate a series from the sequence of its terms.

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.

What it is for: Representing functions, approximating values and solving problems in analysis, physics and engineering.

It can be 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.

Distinguishing features

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 it looks like

Summation notation with a sigma, or terms with plus signs and ellipsis.

How it is recognised

Sigma notation

Named series such as geometric, harmonic, Taylor, Fourier

TV and book series are different senses

Related models

is built from - terms of a sequence

sequence

is confused with - ordered terms, not their sum

sequence

is used in - approximation and expansion

analysis and physics

is confused with - a homonym

series as a set of works

In practice

Families and kinds

geometric and arithmetic series

power, Taylor and Laurent series

Fourier series

formal power series

asymptotic expansions

Standards and regulation

ISO 80000-2 notation

Failure modes and hazards

Summing a divergent series as if convergent

Rearranging conditionally convergent series

Truncating without an error bound

Also called

formal power seriesultrapolynomialquasisymmetric functionasymptotic expansionCornish–Fisher expansionoperator product expansionconvergent seriesmultipole expansionbasic hypergeometric seriesCharlier seriesgeometric seriesfunction seriesLüroth seriescomposition seriesperturbative seriesEdgeworth seriesgeneral Dirichlet seriesHahn seriesinfinite arithmetic seriesPoincaré seriesbinomial seriesdivergent seriesnumerical seriesarithmetic seriesabsolutely convergent seriesconditionally convergent seriesfinite geometric seriesinfinite geometric seriesPuiseux seriesseries diverging to infinity

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.series-cognition

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Terms What is summed.

The terms define the series.

Term rule

General term.

Term rule

The general term.

  1. What is the general term of the series? definition
  2. Where does the summation start? definition

Kind

Geometric, power, Fourier.

Kind

The kind of series.

  1. What kind of series is this? definition
  2. Is it formal or analytic? boundary
Convergence Does it sum.

Convergence decides whether a sum exists.

Tests

Ratio, root, comparison.

Convergence test

Test used.

  1. Does the series converge, by which test? measurement
  2. For which values of the variable? boundary

Type of convergence

Absolute, conditional, uniform.

Convergence type

The type of convergence.

  1. Is the convergence absolute, conditional or uniform? definition
  2. What operations does that permit? boundary
Sum The value.

The sum is the goal when it exists.

Value

Closed form.

Sum

The sum.

  1. What is the sum, in closed form if possible? measurement
  2. How was it derived? provenance

Approximation

Truncation error.

Error bound

Error of partial sums.

  1. How many terms give the needed accuracy? measurement
  2. What bounds the truncation error? measurement
Application Where it is used.

Series are tools for representing functions.

Expansion

Functions as series.

Expansion

Function expansions.

  1. Which function does this series represent, and where? definition
  2. What is the radius of convergence? measurement

Computation

Numerical use.

Numerics

Numerical behaviour.

  1. Is the series numerically stable to sum? boundary
  2. Should acceleration methods be used? action

What the second pass must settle

  • Should named series be separate entries?
  • How should convergence conditions be recorded?
  • How should formal and analytic series be distinguished?
Draft as JSON