Summability Methods
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Home > Art, Film & Photography > Summability Methods: Divergent Series, Tauberian Theorems, Harmonic Series, 1 2 + 3 4 + . . ., Zeta Function Regularization
Summability Methods: Divergent Series, Tauberian Theorems, Harmonic Series, 1 2 + 3 4 + . . ., Zeta Function Regularization

Summability Methods: Divergent Series, Tauberian Theorems, Harmonic Series, 1 2 + 3 4 + . . ., Zeta Function Regularization


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About the Book

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 25. Chapters: Divergent series, Tauberian theorems, Harmonic series, 1 2 + 3 4 + . . ., Zeta function regularization, 1 2 + 4 8 + ..., 1 + 2 + 4 + 8 + ..., Cesaro summation, Borel summation, Series acceleration, Hardy-Littlewood tauberian theorem, Abelian and tauberian theorems, Grandi's series, Ramanujan summation, 1 + 2 + 3 + 4 + ..., Dimensional regularization, Bochner-Riesz mean, Wiener's tauberian theorem, Wiener-Ikehara theorem, 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 Cesaro, Emile 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. Cesaro 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' term...


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Product Details
  • ISBN-13: 9781156676820
  • Publisher: Books LLC, Wiki Series
  • Publisher Imprint: Books LLC, Wiki Series
  • Height: 246 mm
  • No of Pages: 26
  • Spine Width: 1 mm
  • Weight: 68 gr
  • ISBN-10: 1156676827
  • Publisher Date: 26 Jul 2011
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Sub Title: Divergent Series, Tauberian Theorems, Harmonic Series, 1 2 + 3 4 + . . ., Zeta Function Regularization
  • Width: 189 mm


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