We prove a theorem which unifies some formulas, for example the counting function, of some sets of numbers including all positive integers,
-free numbers,
-full numbers, etc. We also establish a ...conjecture and give some examples where the conjecture holds.
It is well-known, as follows from the Stirling’s approximation
, that
. A generalization of this limit is (1
· 2
· · ·
which was established by N. Schaumberger in 1989 (see 8). The aim of this work ...is to establish a new generalization that is in fact an improvement of Schaumberger’s formula for a general sequence
of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.
Let
be the
th prime number. In this note, we study strictly increasing sequences of positive integers
such that the limit lim
(
=
holds. This limit formula is in fact a generalization of some ...previously known results. Furthermore, some other generalizations are established.
In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell ...sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first
terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number
(≈ 2:71828:::).
Notes on a General Sequence Farhadian, Reza; Jakimczuk, Rafael
Annales Mathematicae Silesianae,
09/2020, Letnik:
34, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Let
be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula
, where
are real numbers with
0 and
1. In this note we prove some limits that connect this sequence ...to the number
. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.
We prove general formulae in composition theory. We study the number of restricted compositions of a positive integer
in
parts, where the parts are in very general integer sequences. Note that in ...compositions the order of the parts is considered.
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