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.

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.

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.

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.

scholarly journals Summability of alterations of convergent series

1981 ◽  
Vol 4 (3) ◽  
pp. 543-552
Author(s):  
T. A. Keagy
Keyword(s):  
Convergent Series ◽  
Divergent Series ◽  
Matrix Methods

The effect of splitting, rearrangement, and grouping series alterations on the summability of a convergent series byℓ−ℓandcs−csmatrix methods is studied. Conditions are determined that guarantee the existence of alterations that are transformed into divergent series and into series with preassigned sums.

scholarly journals On the solution of Steinhaus functional equation using weakly Picard operators

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 69-79
Author(s):  
Vasile Berinde
Keyword(s):  
Functional Equation ◽  
Convergent Series ◽  
Existence Results ◽  
Divergent Series ◽  
Picard Operator ◽  
Weakly Picard Operator

In this paper we obtain existence results regarding the solutions g of a Steinhaus type functional equation of the form g(x)+ g(f(x))= F(x), under the significantly weaker assumption that f is a weakly Picard operator. The solutions are given in terms of sums of either convergent series or divergent series but summable by some method of summability.

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