{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.series","code":"thing-q170198","url":"https://ver.cy/models/thing/q170198/","name":"series","alternateNames":["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"],"kind":"thing","status":"research-draft","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.REL"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"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.","purpose":"Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["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"],"structure":{"bundles":[{"id":"terms","name":"Terms","description":"What is summed.","layers":[{"id":"rule","name":"Term rule","description":"General term.","findings":[{"id":"rule-finding","name":"Term rule","description":"The general term.","questions":[{"text":"What is the general term of the series?","kind":"definition"},{"text":"Where does the summation start?","kind":"definition"}]}]},{"id":"kind","name":"Kind","description":"Geometric, power, Fourier.","findings":[{"id":"kind-finding","name":"Kind","description":"The kind of series.","questions":[{"text":"What kind of series is this?","kind":"definition"},{"text":"Is it formal or analytic?","kind":"boundary"}]}]}]},{"id":"convergence","name":"Convergence","description":"Does it sum.","layers":[{"id":"test","name":"Tests","description":"Ratio, root, comparison.","findings":[{"id":"test-finding","name":"Convergence test","description":"Test used.","questions":[{"text":"Does the series converge, by which test?","kind":"measurement"},{"text":"For which values of the variable?","kind":"boundary"}]}]},{"id":"type","name":"Type of convergence","description":"Absolute, conditional, uniform.","findings":[{"id":"type-finding","name":"Convergence type","description":"The type of convergence.","questions":[{"text":"Is the convergence absolute, conditional or uniform?","kind":"definition"},{"text":"What operations does that permit?","kind":"boundary"}]}]}]},{"id":"sum","name":"Sum","description":"The value.","layers":[{"id":"value","name":"Value","description":"Closed form.","findings":[{"id":"value-finding","name":"Sum","description":"The sum.","questions":[{"text":"What is the sum, in closed form if possible?","kind":"measurement"},{"text":"How was it derived?","kind":"provenance"}]}]},{"id":"approximation","name":"Approximation","description":"Truncation error.","findings":[{"id":"approximation-finding","name":"Error bound","description":"Error of partial sums.","questions":[{"text":"How many terms give the needed accuracy?","kind":"measurement"},{"text":"What bounds the truncation error?","kind":"measurement"}]}]}]},{"id":"application","name":"Application","description":"Where it is used.","layers":[{"id":"expansion","name":"Expansion","description":"Functions as series.","findings":[{"id":"expansion-finding","name":"Expansion","description":"Function expansions.","questions":[{"text":"Which function does this series represent, and where?","kind":"definition"},{"text":"What is the radius of convergence?","kind":"measurement"}]}]},{"id":"computation","name":"Computation","description":"Numerical use.","findings":[{"id":"computation-finding","name":"Numerics","description":"Numerical behaviour.","questions":[{"text":"Is the series numerically stable to sum?","kind":"boundary"},{"text":"Should acceleration methods be used?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Determine and state whether a series converges, with the test used.","Truncate a series only with an error bound."],"mustNot":["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."],"requiresHuman":["Relying on a truncated series in safety-critical calculations."]},"ethics":{"considerations":["Mathematical errors in engineering and finance hurt people who trust the result."],"affectedParties":["Users of results","Students"]},"owners":{"steward":"Nobody owns the concept; it is held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q177239","type":"parent","note":"registry parent class"},{"target":"Q218005","type":"parent","note":"registry parent class"},{"target":"Q6029858","type":"parent","note":"registry parent class"},{"target":"sequence","type":"related","note":"terms of a sequence"},{"target":"analysis and physics","type":"related","note":"approximation and expansion"},{"target":"series as a set of works","type":"related","note":"a homonym"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.series","Wikidata: Q170198 (https://www.wikidata.org/wiki/Q170198)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["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":{"applicability":"required","items":["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":{"applicability":"required","items":["Summing a divergent series as if convergent","Rearranging conditionally convergent series","Truncating without an error bound"]},"interfaces":{"applicability":"required","items":["ISO 80000-2 notation"]},"context":{"applicability":"required","items":["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":[{"title":"Wikidata item Q170198: series","url":"https://www.wikidata.org/wiki/Q170198","note":"identity and sense of the item"},{"title":"Wikipedia: Series (mathematics)","url":"https://en.wikipedia.org/wiki/Series_%28mathematics%29","note":"general description of the item"}],"openQuestions":["Should named series be separate entries?","How should convergence conditions be recorded?","How should formal and analytic series be distinguished?"],"resources":{"spec":"/models/things/publications/thing-q170198/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q170198/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T05:00:01Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}