Wyniki wyszukiwania - Biblioteka Nauki

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Wokół liczb i szeregów harmonicznych

100%

Wiśniewski D.

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2015

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tom 7

99-109

EN

The harmonic series is one of the most celebrated infinite series ofmathematics. From a pedagogical point of view, the harmonic series providesa wealth of opportunities. Applications such as Gabriel’s wedding cake andEuler’s proof of the divergence of prime numbers can lead to some verynice discussions. The main idea of this article is to survey some of unusual,insightful and inspiring divergence proofs. First of all, this article is addressedat first-year calculus students.

2

Wokół liczb i szeregów harmonicznych

100%

Wiśniewski D.

|

2015

|

tom 7

99-109

PL

The harmonic series is one of the most celebrated infinite series ofmathematics. From a pedagogical point of view, the harmonic series providesa wealth of opportunities. Applications such as Gabriel’s wedding cake andEuler’s proof of the divergence of prime numbers can lead to some verynice discussions. The main idea of this article is to survey some of unusual,insightful and inspiring divergence proofs. First of all, this article is addressedat first-year calculus students.

3

Composition of arithmetical functions with generalization of perfect and related numbers

75%

Shukla D. P. , Yadav S.

Commentationes Mathematicae

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2012

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tom Vol. 52, [Z] 2

153--170

EN

In this paper we have studied the deficient and abundent numbers connected with the composition of φ,φ*, σ,σ* and ψ arithmetical functions , where φ is the Euler totient, φ* is the unitary totient, σ is the sum of divisors, σ* is the unitary sum of divisors and ip is the Dedekind function. In 1988, J. Sandor conjectured that ψ(φ(m))≥m, for all odd m and proved that this conjecture is equivalent to ψ(φ(m))≥m/2 for all m. Here we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained . Finally, we have discussed the generalization of perfect numbers for an arithmetical function Eα.

4

Harmonic sums and integral representations

51%

Sofo A.

Journal of Applied Analysis

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2010

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tom Vol. 16, nr 2

265-277

EN

In this paper we derive some identities of harmonic number sums with binomial coefficients, we also give integral representations for the sums. We recover some existing identities and introduce a number of new ones.