Related papers: Partial sums of the hyperharmonic series
Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers.…
The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the…
In this paper, we introduce a new type of generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$, and show that Euler sums of the generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$ can be expressed in…
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
Based on some combinatorial identities arising from symbolic summation, we extend two supercongruences on partial sums of hypergeometric series, which were originally conjectured by Guo and Schlosser and recently confirmed by Jana and…
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
A set of integers greater than 1 is primitive if no member in the set divides another. Erd\H{o}s proved in 1935 that the series $f(A) = \sum_{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets $A$. In 1986 he asked…
A famous theorem of Hilbert from 1888 states that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of…
A set of integers greater than 1 is primitive if no element divides another. Erd\H{o}s proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked…
The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and…
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…
In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series.
Inequalities for exponential sums are studied. Our results improve an old result of G. Halasz and a recent result of G. Kos. We prove several other essentially sharp related results in this paper.
In 1946, P. Erd\H{o}s and I. Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1,1 / 2$, $\cdots, 1 / n$ are integers. In 2012, Y. Chen and M. Tang proved…
A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erd\H{o}s showed that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where $\mathscr{A}$ is…
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…
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).
We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb…
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits…