КОМБИНАТОРНОЕ ЗНАЧЕНИЕ ЧИСЛА ЭЙЛЕРА

КОМБИНАТОРНОЕ ЗНАЧЕНИЕ ЧИСЛА ЭЙЛЕРА

Научная статья

Перфилеев М.С. *

ORCID: 0000-0002-9776-3666,

Восточно-Сибирский филиал ФГУП «ВНИИФТРИ», Иркутск, Россия

* Корреспондирующий автор (perfmihserg18011985[at]mail.ru)

Аннотация

В настоящей статье рассматривается новый предел числа e и приводится его научное доказательство с помощью аппарата математического анализа. Используя этот предел, мы проводим комбинаторную интерпретацию числа Эйлера. Это означает, что число Эйлера является отношением количества перестановок (или комбинаций)  на n  к числу перестановок (или комбинаций)   на n с бесконечно большим числом элементов n.

Ключевые слова: число Эйлера, предел функции, формула Стирлинга, число перестановок, количество комбинаций.

COMBINATORIAL MEANING OF EULER'S NUMBER

Research article

Perfileev M.S. *

ORCID: 0000-0002-9776-3666,

East-Siberian branch of FSUE «VNIIFTRI», Irkutsk, Russia

* Corresponding author (perfmihserg18011985[at]mail.ru)

Abstract

In this paper, we consider a new limit for the number e and give its rigorous proof using the apparatus of mathematical analysis. With the help of this limit, a combinatorial interpretation is given for Euler's number. It means that Euler's number is the ratio of the number of permutations (or combinations) of  by n to the number of permutations (or combinations) of  by n with an infinitely large number of elements n.

Keywords: Euler's number, limit of a function, Stirling's formula, number of permutations, number of combinations.

Introduction

The number e is one of the most important mathematical constants [1], and it plays a huge role in differential and integral calculus, in algebra and the laws of physics and chemistry. Euler's number has various representations, for example, sums of infinite series, limits, infinite multiplications, and infinite continued fractions [2], [3].

Let's consider the limit

   (1)

where  p, q, and  r  are real numbers, and  p  is more than zero:  Proof: Let's replace n with  We denote that  . Then we get the limit    (2) Provided  (using Stirling's formula  provided   [5]) we get

(Note: during the transformations, we used the negligible smallness of compared to n provided  and the second remarkable limit   [6]).

Similarly, we can get

Then

Q.E.D.

(Note: during the transformations, we used the negligible smallness of  compared to  provided  and the second remarkable limit).

Combinatorial meaning of Euler's number

Let's consider some properties of the obtained limit (1).

If we take p = 1, and   (Theodorus' constant), and   (Pythagoras' constant) [1], then we get the limit

   (3)

Provided   p = 1, q = 1, r = 0, the limit (1) takes the form of    (4) Replacing n with  we get the expression    (5)

Using highly accurate computer calculations for the expression (5), we can get the following results, presented in this format:

{numeric value n ; deviation of calculated results from the exact value of the number e}:

It is easy to see that as n increases by one order of magnitude, the accuracy of the approximation increases by two orders of magnitude.

Using the formula for the number of permutations of k  elements of m elements,

   [7] can be trivially obtained

   (6)

then Euler's number can be represented as    (7)

Thus, Euler's number also has a combinatorial meaning, which is the ratio of the number of permutations of n elements from  elements to the number of permutations of n elements from   elements with an infinitely large number of elements n.

Also, Euler's number can be interpreted as the ratio of the number of combinations of n elements from  elements to the number of combinations of n elements from  elements with an infinitely large number of elements n.

Indeed, using the formula for the number of combinations

we can obtain    (8) Then we get the result    (9) If in the limit (5) we replace n with  , we get    (10)

(Note: during the transformations, we used the negligible smallness of compared to   provided ). 

Conclusion

Thus, this paper presents and proves a new limit for Euler's number. The properties of this limit have also been considered here. Using the formulas for the number of permutations and combinations of m through k, it is shown that the number e has its combinatorial meaning.

Конфликт интересов Не указан.

Conflict of Interest None declared.

 

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