On convergent and divergent series. Rules for the convergence of series. The summation of several convergent series.

2009 ◽  
pp. 85-115 ◽  
Author(s):  
Robert E. Bradley ◽  
C. Edward Sandifer
Keyword(s):  
Convergent Series ◽  
Divergent Series ◽  
Convergence Of Series

A family of hyper-Bessel functions and convergent series in them

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Bessel Function ◽  
Classical Theory ◽  
Bessel Functions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Convergent Series ◽  
Multi Index ◽  
Fatou Theorem ◽  
Convergence Of Series

AbstractThe Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

scholarly journals VII. A theory of asymptotic series

1912 ◽  
Vol 211 (471-483) ◽  
pp. 279-313 ◽  
Keyword(s):  
Power Series ◽  
Mathematical Analysis ◽  
Asymptotic Series ◽  
Convergent Series ◽  
Divergent Series ◽  
Consistent Theory

In their efforts to place mathematical analysis on the firm est possible foundations, Abel and Cauchy found it necessary to banish non-convergent series from their work ; from that time until a quarter of a century ago the theory o f divergent series was, in general, neglected by mathematicians. A consistent theory of divergent series was, however, developed by Poincaré in 1886, and, ten years later, Borel enunciated his theory of summability in connection with oscillating series. So far as diverging power series are concerned, the theory of Borel is more precise than that of Poincaré.

scholarly journals The Products of Hyperreal Series and the Limitations of Cauchy Products

2021 ◽  
Vol 3 (2) ◽  
pp. 34-36
Author(s):  
Jonathan Bartlett
Keyword(s):  
Convergent Series ◽  
Divergent Series ◽  
Alternative Approaches

Cauchy products are used to take the products of convergent series. Here, we show the limitations of this approach in divergent series, including those that can be analyzed through the BGN method. Alternative approaches and formulas for divergent series are suggested, as well as their benefits and drawbacks.

On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with p-Orthogonal Summands

2001 ◽  
Vol 8 (2) ◽  
pp. 377-388
Author(s):  
Andrew Rosalsky ◽  
Andrei I. Volodin
Keyword(s):  
Banach Space ◽  
Convergent Series ◽  
Difference Sequence ◽  
Martingale Difference ◽  
Nonlinear Anal ◽  
Convergence Of Series ◽  
Special Cases ◽  
Random Elements ◽  
Maximal Moment ◽  
Martingale Convergence

Abstract The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space and for a p-orthogonal sequence of random elements in a Rademacher type p Banach space. The current work generalizes, simplifies, and unifies some of the recent results of Nam and Rosalsky [Teor. Īmovīr. ta Mat. Statist. 52: 120–131, 1995] and Rosalsky and Rosenblatt [Bull. Inst. Math. Acad. Sinica 11: 185–208, 1983, Nonlinear Anal. 30: 4237–4248, 1997].

Rearrangements that Preserve Rates of Divergence

1982 ◽  
Vol 34 (4) ◽  
pp. 916-920
Author(s):  
Elgin H. Johnston
Keyword(s):  
Real Number ◽  
Infinite Series ◽  
Special Interest ◽  
Convergent Series ◽  
Divergent Series ◽  
Classical Theorem ◽  
Real Numbers ◽  
Conditionally Convergent Series ◽  
Positive Integers

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

On the Convergence of a Mapped by a Function

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Peter Eliaš
Keyword(s):  
Divergent Series ◽  
Open Problems ◽  
Convergence Of Series ◽  
Real Functions

AbstractWe provide a characterization of two families of real functions, namely, of those functions f such that the series ∑f(xn) diverges whenever the series ∑xn diverges, or, respectively, whenever the series ∑xn non-absolutely converges. This solves two open problems of J. Borsík. We also reformulate known results on families of functions preserving or changing the type of convergence of series, and add some results about divergent series of terms converging to zero.

Summation of slowly convergent series

1950 ◽  
Vol 46 (3) ◽  
pp. 436-449 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Infinite Series ◽  
Main Part ◽  
Asymptotic Series ◽  
Convergent Series ◽  
Divergent Series ◽  
Finite Form ◽  
Sum Formula ◽  
New Form ◽  
Practical Sense ◽  
Direct Summation

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

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