Matches in SemOpenAlex for { <https://semopenalex.org/work/W1483913247> ?p ?o ?g. }
Showing items 1 to 98 of
98
with 100 items per page.
- W1483913247 abstract "Leonhard Euler (1707–1783) discovered his powerful “summation formula” in the early 1730s. He used it in 1735 to compute the first 20 decimal places for the precise sum of all the reciprocal squares — a number mathematicians had competed to determine ever since the surprising discovery that the alternating sum of reciprocal odd numbers is π/4. This reciprocal squares challenge was called the “Basel Problem”, and Euler achieved his 20-place approximation to it using only a few terms from his diverging summation formula. In contrast, if sought as a simple partial sum of the original slowly converging series, such accuracy would require an overwhelming more than 1020 terms. With his approximation, Euler was probably convincing himself that the sum was exactly π2/6, which spurred his first solution of the celebrated Basel Problem in the same year [7, v. 14, v. 16, section 2, pp. VIIff][18, 26]. We are left in awe that just a few terms of a diverging formula can so closely approximate this sum. Paradoxically, Euler’s formula, even though it usually diverges, provides breathtaking acceleration of approximations for partial and infinite sums of many slowly converging or diverging series. My goal here is to explore Euler’s own mature view of the summation formula and a few of his more diverse applications, largely in his own words from his" @default.
- W1483913247 created "2016-06-24" @default.
- W1483913247 creator A5016126128 @default.
- W1483913247 date "2007-01-01" @default.
- W1483913247 modified "2023-10-16" @default.
- W1483913247 title "Dances between continuous and discrete: Euler’s summation formula" @default.
- W1483913247 cites W149618439 @default.
- W1483913247 cites W1497510637 @default.
- W1483913247 cites W1510232205 @default.
- W1483913247 cites W1519565561 @default.
- W1483913247 cites W153533677 @default.
- W1483913247 cites W1538508619 @default.
- W1483913247 cites W1544888911 @default.
- W1483913247 cites W1554813786 @default.
- W1483913247 cites W1559129531 @default.
- W1483913247 cites W1805871248 @default.
- W1483913247 cites W1986164806 @default.
- W1483913247 cites W1995514749 @default.
- W1483913247 cites W2003640997 @default.
- W1483913247 cites W2171369888 @default.
- W1483913247 cites W2314485382 @default.
- W1483913247 cites W2986789383 @default.
- W1483913247 cites W573545131 @default.
- W1483913247 cites W582836736 @default.
- W1483913247 cites W78118284 @default.
- W1483913247 doi "https://doi.org/10.1090/spec/054/13" @default.
- W1483913247 hasPublicationYear "2007" @default.
- W1483913247 type Work @default.
- W1483913247 sameAs 1483913247 @default.
- W1483913247 citedByCount "1" @default.
- W1483913247 countsByYear W14839132472013 @default.
- W1483913247 crossrefType "other" @default.
- W1483913247 hasAuthorship W1483913247A5016126128 @default.
- W1483913247 hasBestOaLocation W14839132472 @default.
- W1483913247 hasConcept C111472728 @default.
- W1483913247 hasConcept C134306372 @default.
- W1483913247 hasConcept C13626590 @default.
- W1483913247 hasConcept C138885662 @default.
- W1483913247 hasConcept C143724316 @default.
- W1483913247 hasConcept C151730666 @default.
- W1483913247 hasConcept C153363305 @default.
- W1483913247 hasConcept C199343813 @default.
- W1483913247 hasConcept C202444582 @default.
- W1483913247 hasConcept C2777686260 @default.
- W1483913247 hasConcept C2777742833 @default.
- W1483913247 hasConcept C2780586882 @default.
- W1483913247 hasConcept C28826006 @default.
- W1483913247 hasConcept C33923547 @default.
- W1483913247 hasConcept C38409319 @default.
- W1483913247 hasConcept C41895202 @default.
- W1483913247 hasConcept C48330753 @default.
- W1483913247 hasConcept C62884695 @default.
- W1483913247 hasConcept C71924100 @default.
- W1483913247 hasConcept C73905626 @default.
- W1483913247 hasConcept C768646 @default.
- W1483913247 hasConcept C86803240 @default.
- W1483913247 hasConcept C93082080 @default.
- W1483913247 hasConceptScore W1483913247C111472728 @default.
- W1483913247 hasConceptScore W1483913247C134306372 @default.
- W1483913247 hasConceptScore W1483913247C13626590 @default.
- W1483913247 hasConceptScore W1483913247C138885662 @default.
- W1483913247 hasConceptScore W1483913247C143724316 @default.
- W1483913247 hasConceptScore W1483913247C151730666 @default.
- W1483913247 hasConceptScore W1483913247C153363305 @default.
- W1483913247 hasConceptScore W1483913247C199343813 @default.
- W1483913247 hasConceptScore W1483913247C202444582 @default.
- W1483913247 hasConceptScore W1483913247C2777686260 @default.
- W1483913247 hasConceptScore W1483913247C2777742833 @default.
- W1483913247 hasConceptScore W1483913247C2780586882 @default.
- W1483913247 hasConceptScore W1483913247C28826006 @default.
- W1483913247 hasConceptScore W1483913247C33923547 @default.
- W1483913247 hasConceptScore W1483913247C38409319 @default.
- W1483913247 hasConceptScore W1483913247C41895202 @default.
- W1483913247 hasConceptScore W1483913247C48330753 @default.
- W1483913247 hasConceptScore W1483913247C62884695 @default.
- W1483913247 hasConceptScore W1483913247C71924100 @default.
- W1483913247 hasConceptScore W1483913247C73905626 @default.
- W1483913247 hasConceptScore W1483913247C768646 @default.
- W1483913247 hasConceptScore W1483913247C86803240 @default.
- W1483913247 hasConceptScore W1483913247C93082080 @default.
- W1483913247 hasLocation W14839132471 @default.
- W1483913247 hasLocation W14839132472 @default.
- W1483913247 hasOpenAccess W1483913247 @default.
- W1483913247 hasPrimaryLocation W14839132471 @default.
- W1483913247 hasRelatedWork W2028525543 @default.
- W1483913247 hasRelatedWork W2061723820 @default.
- W1483913247 hasRelatedWork W2075653929 @default.
- W1483913247 hasRelatedWork W2081738830 @default.
- W1483913247 hasRelatedWork W2135800810 @default.
- W1483913247 hasRelatedWork W2328313632 @default.
- W1483913247 hasRelatedWork W2355566792 @default.
- W1483913247 hasRelatedWork W2962718271 @default.
- W1483913247 hasRelatedWork W2990993486 @default.
- W1483913247 hasRelatedWork W3006912224 @default.
- W1483913247 isParatext "false" @default.
- W1483913247 isRetracted "false" @default.
- W1483913247 magId "1483913247" @default.
- W1483913247 workType "other" @default.