### $\mathcal{I}$-convergence and extremal $\mathcal{I}$-limit points

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L. Olivier proved in 1827 the classical result about the speed of convergence to zero of the terms of a convergent series with positive and decreasing terms. We prove that this result remains true if we omit the monotonicity of the terms of the series when the limit operation is replaced by the statistical limit, or some generalizations of this concept.

Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived....

An infinite series which arises in certain applications of the Lagrange-Bürmann formula to exponential functions is investigated. Several very exact estimates for the Laplace transform and higher moments of this function are developed.

Using the concept of $\mathcal{I}$-convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.

In this paper we introduce a new sequence space $B{V}_{\sigma}(\mathcal{M},u,p,r,\parallel \xb7,...,\xb7\parallel )$ defined by a sequence of Orlicz functions $\mathcal{M}=\left({M}_{k}\right)$ and study some topological properties of this sequence space.

In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset ${\overline{\mathbb{X}}}_{g}$ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator ${T}_{g}$ to map ${\overline{\mathbb{X}}}_{g}$ into itself or to map ${\mathbb{X}}_{g}$ into...