Summability Theory: Convergence Tests, Divergent Series, Sequence Spaces, Summability Methods, Tauberian Theorems, Harmonic Series

Ön Kapak
General Books, 2011 - 44 sayfa
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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...

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