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Articles 61 - 81 of 81
Full-Text Articles in Mathematics
De Seriebus Divergentibus, Leonhard Euler
De Seriebus Divergentibus, Leonhard Euler
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No abstract provided.
Subsidium Calculi Sinuum, Leonhard Euler
Subsidium Calculi Sinuum, Leonhard Euler
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No abstract provided.
Consideratio Quarumdam Serierum, Quae Singularibus Proprietatibus Sunt Praeditae, Leonhard Euler
Consideratio Quarumdam Serierum, Quae Singularibus Proprietatibus Sunt Praeditae, Leonhard Euler
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No abstract provided.
De Serierum Determinatione Seu Nova Methodus Inveniendi Terminos Generales Serierum, Leonhard Euler
De Serierum Determinatione Seu Nova Methodus Inveniendi Terminos Generales Serierum, Leonhard Euler
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No abstract provided.
Methodus Facilis Computandi Angulorum Sinus Ac Tangentes Tam Naturales Quam Artificiales, Leonhard Euler
Methodus Facilis Computandi Angulorum Sinus Ac Tangentes Tam Naturales Quam Artificiales, Leonhard Euler
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No abstract provided.
De Fractionibus Continuis Observationes, Leonhard Euler
De Fractionibus Continuis Observationes, Leonhard Euler
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No abstract provided.
De Productis Ex Infinitis Factoribus Ortis, Leonhard Euler
De Productis Ex Infinitis Factoribus Ortis, Leonhard Euler
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No abstract provided.
Consideratio Progressionis Cuiusdam Ad Circuli Quadraturam Inveniendam Idoneae, Leonhard Euler
Consideratio Progressionis Cuiusdam Ad Circuli Quadraturam Inveniendam Idoneae, Leonhard Euler
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No abstract provided.
De Seriebus Quibusdam Considerationes, Leonhard Euler
De Seriebus Quibusdam Considerationes, Leonhard Euler
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No abstract provided.
Variae Observationes Circa Series Infinitas, Leonhard Euler
Variae Observationes Circa Series Infinitas, Leonhard Euler
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No abstract provided.
De Fractionibus Continuis Dissertatio, Leonhard Euler
De Fractionibus Continuis Dissertatio, Leonhard Euler
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No abstract provided.
De Variis Modis Circuli Quadraturam Numeris Proxime Exprimendi, Leonhard Euler
De Variis Modis Circuli Quadraturam Numeris Proxime Exprimendi, Leonhard Euler
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No abstract provided.
Démonstration De La Somme De Cette Suite 1 + 1/4 + 1/9 + 1/16 + ..., Leonhard Euler
Démonstration De La Somme De Cette Suite 1 + 1/4 + 1/9 + 1/16 + ..., Leonhard Euler
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No abstract provided.
Inventio Summae Cuiusque Seriei Ex Dato Termino Generali, Leonhard Euler
Inventio Summae Cuiusque Seriei Ex Dato Termino Generali, Leonhard Euler
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Euler continues with the methods of E25 to attack ζ(2) for a second time. He starts with a Taylor series, builds a "Bernoulli polynomial" and uses it to evaluate 0n + 1n + 2n + 3n + ... + (x-1)n, (x = 1, 2, 3, ...) and gets the relationship (B+1)n+1 – Bn+1 = 0 for Bernoulli numbers. He gets an infinite series approximation for the nth partial sum of the harmonic series.
Methodus Universalis Series Summandi Ulterius Promota, Leonhard Euler
Methodus Universalis Series Summandi Ulterius Promota, Leonhard Euler
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No abstract provided.
Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, Leonhard Euler
Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, Leonhard Euler
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This paper begins with an expression of the trapezoid rule for "formal mechanical quadrature." Euler also sums the first ten terms of ζ(2) to get 1.549768 and gives an expression for the error term. Then he determines the sum of the first million terms of the harmonic series to be 14.392669.
De Summis Serierum Reciprocarum, Leonhard Euler
De Summis Serierum Reciprocarum, Leonhard Euler
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Euler finds the values of ζ(2n), where ζ is now named as the Riemann zeta function. He introduces the Euler numbers En. All the odd-indexed Euler numbers are zero, E0=1, E2=-1, E4=5, E6=-61, E8=1380, and they satisfy (E+1)n+(E-1)n = 0. (The Euler numbers are the coefficients of the series expansion of sech(x) and are related to Bernoulli numbers.) Among other things, this paper includes an infinite product formula for sin(x)/x.
De Progressionibus Harmonicis Observationes, Leonhard Euler
De Progressionibus Harmonicis Observationes, Leonhard Euler
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Euler gives the now-named Euler-Mascheroni constant, accurate to five decimal places, and examines several series related to log(n).
De Summatione Innumerabilium Progressionum, Leonhard Euler
De Summatione Innumerabilium Progressionum, Leonhard Euler
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This paper concerns the sum of reciprocal squares, which equals π2/6. Euler does not yet have the tools to find this value directly, but instead approximates it as 1.644934. He says this follows from E25 and E19, and also refers us forward to E736. Then Euler brings in the harmonic series: letting f(x) denote the xth partial sum of the harmonic series, he approximates it as an integral and defines his constant γ as the limit of f(x) – log(x).
De Progressionibus Transcendentibus Seu Quarum Termini Generales Algebraice Dari Nequeunt, Leonhard Euler
De Progressionibus Transcendentibus Seu Quarum Termini Generales Algebraice Dari Nequeunt, Leonhard Euler
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No abstract provided.
Methodus Generalis Summandi Progressiones, Leonhard Euler
Methodus Generalis Summandi Progressiones, Leonhard Euler
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This paper leads to Bernoulli numbers from an integral of an infinite series and is called a beautiful triumph by Euler.