Euler’s method of rearranging absolutely convergent series
E300760
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Euler’s method of rearranging absolutely convergent series canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815442 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Euler’s method of rearranging absolutely convergent series Context triple: [Euler product formula for the Riemann zeta function, dependsOn, Euler’s method of rearranging absolutely convergent series]
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A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
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B.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
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C.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
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D.
Halley’s method for solving equations
Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
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E.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Euler’s method of rearranging absolutely convergent series Target entity description: Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
-
C.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
-
D.
Halley’s method for solving equations
Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
-
E.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
- F. None of above. chosen
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical method
ⓘ
method for manipulating infinite series ⓘ technique in analysis ⓘ |
| appliesTo |
absolutely convergent series
ⓘ
infinite series ⓘ |
| approach | systematic reindexing and regrouping of terms in a series ⓘ |
| assumes | the series under consideration is absolutely convergent ⓘ |
| basedOn | absolute convergence of series ⓘ |
| category | Eulerian method ⓘ |
| context |
classical theory of infinite series
ⓘ
foundations of analytic number theory ⓘ |
| contrastWith | rearrangements of conditionally convergent series ⓘ |
| field |
analytic number theory
ⓘ
mathematical analysis ⓘ |
| guarantees | the sum of the series is unchanged by rearrangement ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| influenced |
development of Euler products for L-functions
ⓘ
later techniques in analytic number theory ⓘ |
| introducedBy | Leonhard Euler ⓘ |
| namedAfter | Leonhard Euler ⓘ |
| propertyUsed | rearrangements of absolutely convergent series preserve the sum ⓘ |
| purpose |
to derive new series identities
ⓘ
to obtain product expansions ⓘ to systematically reorder convergent infinite series ⓘ |
| relatedTo |
Dirichlet series
ⓘ
Euler products for automorphic L-functions ⓘ
surface form:
Euler product expansions
rearrangement theorem for absolutely convergent series ⓘ series acceleration techniques ⓘ zeta function expansions ⓘ |
| requires | control over convergence of partial sums ⓘ |
| usedFor |
deriving identities in analytic number theory
ⓘ
deriving identities involving special functions ⓘ transforming series into product forms ⓘ |
| usedIn |
derivations of product formulas for the sine function
ⓘ
derivations of product formulas for trigonometric functions ⓘ manipulation of power series expansions ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Euler’s method of rearranging absolutely convergent series Description of subject: Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Euler product formula for the Riemann zeta function
→
dependsOn
→
Euler’s method of rearranging absolutely convergent series
ⓘ