Ramanujan Summation Of Divergent Series

Author by : Bernard Candelpergher
Language : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 69
Total Download : 896
File Size : 44,7 Mb
pdf pdf

Description : The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent series. Several examples and applications are given. For numerical evaluation, a formula in terms of convergent series is provided by the use of Newton interpolation. The relation with other summation processes such as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic theory is developed that unifies all these summation processes. This monograph is aimed at graduate students and researchers who have a basic knowledge of analytic function theory.


Divergent Series

Author by : Books Llc
Language : en
Publisher by : Books LLC, Wiki Series
Format Available : PDF, ePub, Mobi
Total Read : 61
Total Download : 738
File Size : 42,5 Mb
pdf pdf

Description : Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 23. Chapters: Summability methods, Harmonic series, 1 2 + 3 4 +, Zeta function regularization, 1 2 + 4 8 + ..., 1 + 2 + 4 + 8 + ..., Ces ro summation, Borel summation, Series acceleration, Abelian and tauberian theorems, Grandi's series, Ramanujan summation, 1 + 2 + 3 + 4 + ..., Dimensional regularization, Bochner-Riesz mean, Divergent geometric series, Hadamard regularization, 1 + 1 + 1 + 1 + ..., Lambert summation, Silverman-Toeplitz theorem, Resummation. Excerpt: In mathematics, 1 2 + 3 4 + is the infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as The infinite series diverges, meaning that its sequence of partial sums, does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation: A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Ces ro, mile Borel and others investigated well-defined methods to assign generalized sums to divergent series-including new interpretations of Euler's attempts. Many of these summability methods easily assign to a "sum" of after all. Ces ro summation is one of the few methods that do not sum, so the series is an example where a slightly stronger method, such as Abel summation, is required. The series 1 2 + 3 4 + ... is closely related to Grandi's series . Euler treated these two as special cases of for arbitrary n, a line of research extending his work on the Basel problem and leading towards the functional equations of what we now know as the Dirichlet eta function and the Riemann zeta function. The series' terms (1, 2, 3, 4, ...) do not approach 0; therefore diverges by the term test. For later reference, it wil...


Summability Theory

Author by : Source: Wikipedia
Language : en
Publisher by : Books LLC, Wiki Series
Format Available : PDF, ePub, Mobi
Total Read : 39
Total Download : 588
File Size : 53,7 Mb
pdf pdf

Description : Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 42. Chapters: Convergence tests, Divergent series, Sequence spaces, Summability methods, Tauberian theorems, Harmonic series, 1 ? 2 + 3 ? 4 +, Riemann series theorem, Absolute convergence, Zeta function regularization, Ratio test, 1 ? 2 + 4 ? 8 + ..., 1 + 2 + 4 + 8 + ..., Ces ro summation, Integral test for convergence, Borel summation, Series acceleration, Hardy-Littlewood tauberian theorem, Abelian and tauberian theorems, Root test, Grandi's series, Ramanujan summation, 1 + 2 + 3 + 4 + ..., Alternating series test, Dimensional regularization, Bochner-Riesz mean, Cauchy condensation test, Abel's test, Term test, Weierstrass M-test, Wiener's tauberian theorem, Mertens' theorems, Wiener-Ikehara theorem, Dini test, Dirichlet's test, Divergent geometric series, Hadamard regularization, 1 + 1 + 1 + 1 + ..., Comparison test, Stolz-Ces ro theorem, Lambert summation, Conditional convergence, Abel's uniform convergence test, Unconditional convergence, Cauchy's convergence test, Retkes convergence criterion, Silverman-Toeplitz theorem, Resummation. Excerpt: In mathematics, 1 ? 2 + 3 ? 4 + is the infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as The infinite series diverges, meaning that its sequence of partial sums, does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation: A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Ces ro, mile Borel and others investigated well-defined methods to assign generalized sums to divergent series-including new interpretations of Euler's attempts. Many of these summability methods easily assign to a "sum" of after all. Ces ro summa...


Illuminism Contra Discordianism

Author by : Brother Cato
Language : en
Publisher by : Lulu Press, Inc
Format Available : PDF, ePub, Mobi
Total Read : 97
Total Download : 362
File Size : 55,8 Mb
pdf pdf

Description : “Philosophy is a battle against the bewitchment of our intelligence by means of language.” – Wittgenstein. We use language to think, to talk to each other, to write, to form beliefs, religions, philosophies, and so on. But what if we are using the wrong language? Do we have “wrong thoughts” because we are using the wrong language? Do we have wrong religions and philosophies for exactly the same reason? What’s the right language? If we could find the right language, could we then think correctly, without error, without delusion, without fantasy? Would the right language give us the right religion, the right philosophy? Would it explain reality to us? Discordians hate Truth. The struggle between the Discordians (in all their various forms) and the Illuminati is the most important there is. The soul of humanity is at stake. The Truth itself is the prize to be won or lost.


Ramanujan

Author by : Srinivasa Ramanujan Aiyangar
Language : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 88
Total Download : 227
File Size : 55,5 Mb
pdf pdf

Description : The letters that Ramanujan wrote to G. H. Hardy on January 16 and February 27, 1913, are two of the most famous letters in the history of mathematics. These and other letters introduced Ramanujan and his remarkable theorems to the world and stimulated much research, especially in the 1920s and 1930s. This book brings together many letters to, from, and about Ramanujan. The letters came from the National Archives in Delhi, the Archives in the State of Tamil Nadu, and a variety of other sources. Helping to orient the reader is the extensive commentary, both mathematical and cultural, by Berndt and Rankin; in particular, they discuss in detail the history, up to the present day, of each mathematical result in the letters. Containing many letters that have never been published before, this book will appeal to those interested in Ramanujan's mathematics as well as those wanting to learn more about the personal side of his life. Ramanujan: Letters and Commentary was selected for the CHOICE list of Outstanding Academic Books for 1996.


Crc Standard Mathematical Tables And Formulas 33rd Edition

Author by : Daniel Zwillinger
Language : en
Publisher by : CRC Press
Format Available : PDF, ePub, Mobi
Total Read : 20
Total Download : 685
File Size : 47,8 Mb
pdf pdf

Description : Containing more than 6,000 entries, CRC Standard Mathematical Tables and Formulas, 33rd Edition continues to provide essential formulas, tables, figures and detailed descriptions. The newest edition of this popular series also features many diagrams, group tables, and integrals that are not available online. This edition also incorporates important topics such as max plus algebra, financial options, pseudospectra, and proof methods. Newly updated topics reflecting new results include coupled analogues, general relativity, radar, and significant equations of mathematics.


Crc Standard Mathematical Tables And Formulas 33rd Edition

Author by : Daniel Zwillinger
Language : en
Publisher by : CRC Press
Format Available : PDF, ePub, Mobi
Total Read : 97
Total Download : 164
File Size : 46,9 Mb
pdf pdf

Description : Containing more than 6,000 entries, CRC Standard Mathematical Tables and Formulas, 33rd Edition continues to provide essential formulas, tables, figures and detailed descriptions. The newest edition of this popular series also features many diagrams, group tables, and integrals that are not available online. This edition also incorporates important topics such as max plus algebra, financial options, pseudospectra, and proof methods. Newly updated topics reflecting new results include coupled analogues, general relativity, radar, and significant equations of mathematics.


Mathematical Series

Author by : Source Wikipedia
Language : en
Publisher by : University-Press.org
Format Available : PDF, ePub, Mobi
Total Read : 32
Total Download : 191
File Size : 46,6 Mb
pdf pdf

Description : Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 133. Chapters: Taylor series, Pi, Geometric series, Taylor's theorem, Power series, Uniform convergence, Formal power series, Laurent series, Lp space, Harmonic series, Arithmetic progression, Monotone convergence theorem, History of Grandi's series, Hypergeometric function, Madhava series, Generalized hypergeometric function, 1 2 + 3 4 + . . ., Sylvester's sequence, Riemann series theorem, Geometric progression, Radius of convergence, Divergent series, Absolute convergence, Summation of Grandi's series, Appell series, Occurrences of Grandi's series, The sum of the reciprocals of the primes diverges, Eisenstein series, Convergent series, Dirichlet series, Volterra series, List of mathematical series, Binomial series, Ratio test, 1/4 + 1/16 + 1/64 + 1/256 + . . ., 1 2 + 4 8 + ..., Edgeworth series, General Dirichlet series, 1 + 2 + 4 + 8 + ..., Kempner series, Asymptotic theory, Cesaro summation, Integral test for convergence, Lauricella hypergeometric series, Borel summation, Charles Haros, Telescoping series, Asymptotic expansion, Abelian and tauberian theorems, Alternating series, Convergence tests, Grandi's series in education, Evaluating sums, Root test, Ramanujan summation, 1 + 2 + 3 + 4 + ..., Alternating series test, Cauchy-Hadamard theorem, Summation by parts, Small set, Problems involving arithmetic progressions, Hilbert-Poincare series, Cauchy condensation test, Abel's test, Abel's theorem, Goldbach-Euler theorem, Weierstrass M-test, Lambert series, Humbert series, Sequence transformation, Spectrum continuation analysis, Mercator series, Van Wijngaarden transformation, Kolmogorov's three-series theorem, Mertens' theorems, 1/2 1/4 + 1/8 1/16 + . . ., Retkes identities, Euler summation, Dirichlet's test, Divergent geometric series, Cesaro mean, Sturm series, Wiener series, Neumann...


Theory And Applications Of Special Functions

Author by : Mourad E. H. Ismail
Language : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 70
Total Download : 719
File Size : 51,5 Mb
pdf pdf

Description : A collection of articles on various aspects of q-series and special functions dedicated to Mizan Rahman. It also includes an article by Askey, Ismail, and Koelink on Rahman’s mathematical contributions and how they influenced the recent upsurge in the subject.


Mathematics Magazine

Author by :
Language : en
Publisher by :
Format Available : PDF, ePub, Mobi
Total Read : 56
Total Download : 344
File Size : 52,5 Mb
pdf pdf

Description :