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.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Achievement sets and sum ranges with ideal supports

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4911-4922
Author(s):  
Jacek Marchwicki
Keyword(s):  
Convergent Series ◽  
Topological Properties ◽  
Real Numbers ◽  
Small Set ◽  
Conditionally Convergent Series

We introduce the notion of ideally supported achievement sets for a series of real numbers. We analize their complexity and topological properties. We compare the notion of ideal achievement sets with the notion of ideally supported sum range of real series, considered by Filip?w and Szuca. We complete Filip?w and Szuca characterization of ideal sum ranges, [R. Filip?w, P. Szuca, Rearrangement of conditionally convergent series on a small set, J. Math. Anal. Appl. 362 (2010), no. 1, 64-71.], and we obtain some generalization of Riemann?s Theorem.

scholarly journals Generalized ideal convergence in intuitionistic fuzzy normed linear spaces

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 811-820 ◽  
Author(s):  
Bipan Hazarika ◽  
Vijay Kumar ◽  
Bernardo Lafuerza-Guilién
Keyword(s):  
Real Number ◽  
Normed Spaces ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Linear Spaces ◽  
Intuitionistic Fuzzy ◽  
Cauchy Sequences ◽  
Positive Integers ◽  
Intuitionistic Fuzzy Normed Spaces ◽  
Cluster Points

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In [19], Kostyrko et al. introduced the concept of ideal convergence as a sequence (xk) of real numbers is said to be I-convergent to a real number e, if for each ? > 0 the set {k ? N : |xk - e| ? ?} belongs to I. The aim of this paper is to introduce and study the notion of ?-ideal convergence in intuitionistic fuzzy normed spaces as a variant of the notion of ideal convergence. Also I? -limit points and I?-cluster points have been defined and the relation between them has been established. Furthermore, Cauchy and I?-Cauchy sequences are introduced and studied. .

scholarly journals Ramanujan Summation for Powers of Triangular and Pronic Numbers

2020 ◽  
Vol 5 (1-2) ◽  
pp. 05-08
Author(s):  
Dr. R. Sivaraman
Keyword(s):  
Divergent Series ◽  
Natural Numbers ◽  
Real Numbers ◽  
Triangular Numbers ◽  
Positive Integers

The numbers which are sum of first n natural numbers are called Triangular numbers and numbers which are product of two consecutive positive integers are called Pronic numbers. The concept of Ramanujan summation has been dealt by Srinivasa Ramanujan for divergent series of real numbers. In this paper, I will determine the Ramanujan summation for positive integral powers of triangular and Pronic numbers and derive a new compact formula for general case.

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.

On Some Diophantine Inequalities Involving the Exponential Function

1965 ◽  
Vol 17 ◽  
pp. 616-626 ◽  
Author(s):  
A. Baker
Keyword(s):  
Real Number ◽  
Exponential Function ◽  
Diophantine Inequalities ◽  
Real Numbers ◽  
Image Position ◽  
Positive Integers

It is well known that for any real number θ there are infinitely many positive integers n such thatHere ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

scholarly journals On generalized difference ideal convergence in random 2-normed spaces

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1273-1282 ◽  
Author(s):  
Bipan Hazarika
Keyword(s):  
Real Number ◽  
Normed Spaces ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Cauchy Sequences ◽  
Convergence Of Sequences ◽  
Positive Integers

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In [17], Kostyrko et. al introduced the concept of ideal convergence as a sequence (xk ) of real numbers is said to be I-convergent to a real number ?, if for each ? > 0 the set {k ? N : |xk ? ?| ? ?} belongs to I. In [28], Mursaleen and Alotaibi introduced the concept of I-convergence of sequences in random 2-normed spaces. In this paper, we define and study the notion of ?n -ideal convergence and ?n -ideal Cauchy sequences in random 2-normed spaces, and prove some interesting theorems.

scholarly journals Some Series and Mathematic Constants Arising in Radioactive Decay

2019 ◽  
Vol 11 (6) ◽  
pp. 14
Author(s):  
Xun Zhou
Keyword(s):  
Infinite Series ◽  
Half Life ◽  
Radioactive Decay ◽  
Convergent Series ◽  
Divergent Series ◽  
Radioactive Isotopes ◽  
Initial Value ◽  
Related Series ◽  
Mathematical Relations ◽  
Mathematical Constants

In this paper we show the construction of 32 infinite series based on the law of decay of radioactive isotopes, which indicates that a radioactive parent isotope is reduced by 1/2 and 1/e of its initial value during each half-life and mean life, respectively. We found that the ratios among the values of the radioactive parent isotope and the radiogenic daughter isotope for each half-life&rsquo;s and mean life&rsquo;s decay can be used to construct 16 half-life related (or 2-related) and 16 mean life related (or e-related) infinite series. There are 8 divergent series, 4 previously known convergent series and 2 series converging to the Erd&ouml;s-Borwein constant. The remaining 18 series are found to converge to 18 mathematical constants and the divergent and alternating mean life related series leads to another 2 mathematical constants. A few interesting mathematical relations exist among these convergent series and 5 sequences are also attained from the convergent half-life related series.

scholarly journals SEL Series Expansion and Generalized Model Construction for the Real Number System via Series of Rationals

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Patiwat Singthongla ◽  
Narakorn Rompurk Kanasri
Keyword(s):  
Real Number ◽  
Series Expansion ◽  
Infinite Series ◽  
Number System ◽  
Model Construction ◽  
Series Expansions ◽  
Real Numbers ◽  
Generalized Model ◽  
The Real ◽  
Real Number System

We present an algorithm for constructing infinite series expansion for real numbers, which yields generalized versions of three famous series expansions, namely, Sylvester series, Engel series, and Lüroth series expansions. Using series of rationals, a generalized model for the real number system is also constructed.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

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