相关论文: Evaluating the sum of convergent positive series

We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.

From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.

Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of…

By means of the notion of umbrae indexed by multisets, a general method to express estimators and their products in terms of power sums is derived. A connection between the notion of multiset and integer partition leads immediately to a way…

In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.

We study complete convergence and closely related Hsu-Robbins-Erd\H{o}s-Spitzer-Baum-Katz series for sums whose terms are elements of linear autoregression sequences. We obtain criterions for convergence of this series expressed in moment…

We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such…

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space…

In this paper, we provide formulas for partial sums of weighted averages over regular integers modulo $n$ of the $\gcd$-sum function with any arithmetic function. Many interesting applications of the results are also given.

A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the knowledge of the exact asymptotic parameters. The method is…

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers…

Methods of determination of constants of the Standard Model are considered. The constants values obtained now are presented and experiments for improving some values are pointed out. A few possible generalized models are considered together…

We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.

The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function.

The higher criticism of a family of tests starts with the individual uncorrected p-values of each test. It then requires a procedure for deciding whether the collection of p-values indicates the presence of a real effect and if possible…

A general method to combine several estimators of the same quantity is investigated. In the spirit of model and forecast averaging, the final estimator is computed as a weighted average of the initial ones, where the weights are constrained…

We study the concept of universal sets from the additive--combinatorial point of view. Among other results we obtain some applications of this type of uniformity to sets avoiding solutions to linear equations, and get an optimal upper bound…