Results 291 to 300 of about 640,777 (319)

Summation of Harmonic Numbers

1989
The problem of finding closed forms for a summation involving harmonic numbers is considered. Solutions for ∑ i n =1P(i)H i (k) , where p(i) is a polynomial, and ∑ i n =1 Hi/(i+m), where m is an integer, are given. A method to automate these results is presented.
E. A. Lamagna, D. Y. Savio, S. M. Liu
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Inequalities for the harmonic numbers

Mathematische Zeitschrift, 2009
We present various inequalities for the harmonic numbers defined by \({H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}\). One of our results states that we have for all integers n ≥ 2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\
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Identities on harmonic and q-harmonic number sums

Afrika Matematika, 2011
By partial fraction approach, we derive q-analog for several well known results on harmonic number sums.
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Harmonic Numbers of Any Order and the Wolstenholme’s-Type Relations for Harmonic Numbers

2016
The concept of harmonic numbers has appeared permanently in the mathematical science since the very early days of differential and integral calculus. Firsts significant identities concerning the harmonic numbers have been developed by Euler (see Basu, Ramanujan J, 16:7–24, 2008, [1], Borwein and Bradley, Int J Number Theory, 2:65–103, 2006, [2], Sofo ...
Mariusz Pleszczyński, Piotr Lorenc, M. Szweda, Michał Różański, Edyta Hetmaniok, Roman Wituła   +5 more
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Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Applied Mathematics and Computation, 2011
Abstract Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics.
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On the Ramanujan Harmonic Number Expansion [PDF]

Results in Mathematics, 2018
In this paper, we give a recursive relation for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number, without the Bernoulli numbers and polynomials.
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Ramanujan’s formula for the harmonic number

Applied Mathematics and Computation, 2018
In this paper, we investigate certain asymptotic series used by Hirschhorn to prove an asymptotic expansion of Ramanujan for the nth harmonic number. We give a general form of these series with a recursive formula for its coefficients. By using the result obtained, we present a formula for determining the coefficients of Ramanujans asymptotic expansion
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10368: A Harmonic-Perfect Number

This report proposes a new mathematical classification: the Harmonic-Perfect Number (HPN). The number 10368 is examined in detail and shown to exhibit harmonic, geometric, and arithmetic resonance properties—including triangular number status, base-12 symmetry, rich divisor structure, and musical interval correspondence.
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Cancer treatment and survivorship statistics, 2022

Ca-A Cancer Journal for Clinicians, 2022
Kimberly D Miller, Leticia M Nogueira, Angela B Mariotto   +2 more
exaly  

Proportion and number of cancer cases and deaths attributable to potentially modifiable risk factors in the United States

Ca-A Cancer Journal for Clinicians, 2018
Farhad Islami, Ann Goding Sauer Msph, Kimberly D Miller   +2 more
exaly