Gold trader. The Relationship Between Even Powers of the Gaussian Integral, the Gamma Function, and Fibonacci Levels

Write from the position of the bundle of the number Pi

Let us consider the number π from the perspective of its various mathematical decompositions («bundles») – an approach that allows us to see the internal structure of the number, its representations in different mathematical contexts and its connection with fundamental functions and formulas.

1. Decimal (and other positional) expansion

The most obvious stratification is the expansion of a number into powers of the base of the number system:

The same is possible in other systems (for example, binary), where each digit gives its own «layer» of the number [20].

2. Continued fraction expansion

The number π can be represented as a continued fraction, which reveals its layered approximate structure:

This series is important because it gives the best possible approximation by a number (e.g. 22/7, 333/106, etc.) [20] [21].

3. Expansions through infinite series

There are many expansions of π in the form of infinite series. For example, Leibniz’s formula:

or Gregory’s formula:

Of the same type are the formulas of Ramanujan, Chudnovsky, and others, which often include factorials, radicals, and Gamma function numbers [22] [23].

4. Stratification through integrals and associated functions

π can be defined as the value of a definite integral, which leads to its representation through a «layered» structure of functions: