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 |
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical method
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method for manipulating infinite series ⓘ technique in analysis ⓘ |
| appliesTo |
absolutely convergent series
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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
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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
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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
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derivations of product formulas for trigonometric functions ⓘ manipulation of power series expansions ⓘ |
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
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dependsOn
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Euler’s method of rearranging absolutely convergent series
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