ABOUT CONVERGENCE OF THE GENERALIZED HARMONIC SERIES WHEN THE SIGNS OF ITS SUMMANDS ARE CHANGED
Journal Title: Дослідження в математиці і механіці - Year 2018, Vol 23, Issue 1
Abstract
In this paper we study the question about convergence of the generalized harmonic series when the signs of its summands are changed. For this we consider the sequence of numbers where the sign is switched. The main result is the convergence criterion of a generalized harmonic series with changed signs in terms of the sign switching numbers. This criterion allows us to reduce the question of convergence to the study of a simpler, namely, sign alternating series. There are given several examples of the application of the obtained criterion. The most interesting was the example of a power growth of the sign switching numbers, because there is a transition to integer parts in the case when the exponent is not natural. Necessary and sufficient conditions for convergence are obtained, but the question is not solved in full, because these conditions do not coincide.
Authors and Affiliations
S. R. Voronkova
Keywords
Related Articles
- EP ID EP558743
- DOI 10.18524/2519-206x.2018.1(31).134617
- Views 105
- Downloads 0
Simultaneous approximation of locally integrable functions and their ψ-integrals
Low smoothness case. The article presents the problems of simultaneous approximation of locally integrable functions on the real axis of low smoothness and their integrals using Vallee Poussin operators. The asymptotic l...
Hyperbolic equation with singular coefficient
Dedicated to research existence conditions of quasi-linear differential equations with measurable coefficients, ie the study limitations imposed by the non linearity in which the system will have to research a solution a...
About additive operation on set of discrete random variable
Authors propose an approach to determine an additive operation on set of discrete random variables. Result of this operation has as much possible values, how many operands. Conditions under which operation is commutative...
One generalization of the classical orthogonal polynomials
The differential equation of the second order, generalizing the differential equations leaded to Jacobi, Laguerre and Hermite polynomials, is considered in the paper. The orthogonality of the polynomials, which are the s...
OSCILLATION CRITERIA FOR HIGHER ORDER SUBLINEAR DELAY DIFFERENTIAL EQUATIONS
In the interval [a,+∞[, the sublinear differential equations of order n⩾2 are considered. A solution of such equation is called proper if it is not identically equal to zero in any neighbourhood of +∞. The proper solutio...