On the Convergence of a Mapped by a Function
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Peter Eliaš
Abstract
We 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.
References
[1] BORSÍK, J.: Functions preserving some types of series, J. Appl. Anal. 14 (2008), 149-163.10.1515/JAA.2008.149Search in Google Scholar
[2] BORSÍK, J.-ČERVEˇNANSK´Y, J.-ŠALÁT, T.: Remarks on functions preserving convergence of infinite series, Real Anal. Exchange 21 (1995/96), 725-731.10.2307/44152683Search in Google Scholar
[3] RADO R.: A theorem on infinite series, J. Lond. Math. Soc. (2) 35 (1960), 273-276. Search in Google Scholar
© 2015 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- The M-Regular Graph of a Commutative Ring
- Further Properties of the Lattice of Torsion Classes of Abelian Cyclically Ordered Groups
- Free Power-Commutative Groupoids
- On Systems of Independent Sets
- Note on G-Module Structure of Orders
- Semisimple Hopf Algebras of Dimension 2q3
- On the Convergence of a Mapped by a Function
- Multiple Solutions of Nonlinear Fractional Differential Equations with p-Laplacian Operator and Nonlinear Boundary Conditions
- Meromorphic Functions Sharing Pairs of Small Functions
- Sharp Inequalities for Polygamma Functions
- Existence Results for Some Higher-Order Evolution Equations with Time-Dependent Unbounded Operator Coefficients
- Existence of Ground State Solutions for Hamiltonian Elliptic Systems with Gradient Terms
- Approximate Higher Ring Derivations in Non-Archimedean Banach Algebras
- Characterizations of Banach Spaces which are not Isomorphic to any of their Proper Subspaces
- Operator Inequalities Related to Q-Class Functions
- On a Class of Locally Dually Flat (α,β)–Metrics
- Limit Theorems for the Counting Function of Eigenvalues up to Edge in Covariance Matrices
- A Generalization of Jakubec's Formula
