Related papers: The Harmonic Series and the nth Term Test for Dive…
The harmonic numbers $H_n=\sum_{0<k\le n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences:…
In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results.
During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously…
We provide numerical procedures for possibly best evaluating the sum of positive series. Our procedures are based on the application of a generalized version of Kummer's test.
In this paper we prove that the weighted linear combination of products of the k-subsets of an n-set of positive real numbers with weight being the harmonic mean of their reciprocal sets is less than or equal to uniformly weighted sum of…
The main objective of this paper is to investigate the explicit form, stability character and global behavior of solutions of the following two systems of rational difference equations x_{n+1}=((+(-)1)/(y_{n}(x_{n-1}+(-)1)+1)),…
In this paper we discuss some convergence and divergence properties of subsequences of logarithmic means of Walsh-Fourier series . We give necessary and sufficient conditions for the convergence regarding logarithmic variation of numbers.
Boson sampling is one of the leading protocols for demonstrating a quantum advantage, but the theory of how this protocol responds to noise is still incomplete. We extend the theory of classical simulation of boson sampling with partial…
Usually one compares the accuracy of two competing classifiers via null hypothesis significance tests (nhst). Yet the nhst tests suffer from important shortcomings, which can be overcome by switching to Bayesian hypothesis testing. We…
The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonously" convergent to zero.…
The slowly converging series sum_{k=3}^infinity 1/[k * log k * (log log k)^a] is evaluated to 38.4067680928 at a=2. After some initial terms, the infinite tail of the sum is replaced by the integral of the associated interpolating function,…
We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…
In this paper, we obtain some formulae for harmonic sums, alternating harmonic sums and Stirling number sums by using the method of integral representations of series. As applications of these formulae, we give explicit formula of several…
Most music theory books are like medieval medical textbooks: they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases. How does music, in particular harmony, actually work, presented as a real,…
Hypothesis testing in high dimensional data is a notoriously difficult problem without direct access to competing models' likelihood functions. This paper argues that statistical divergences can be used to quantify the difference between…
We present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) Hurwitz zeta functions, which allows a…
This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for…
The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers $\sum_{n=0}^{\infty}a_n$ are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for…
The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose…