A Geometric Conception of Euler’s constant
Supported by Newton’s Binomial
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The mathematical constant that defines the base of natural logarithms is Euler’s constant. Its theoretical value corresponds to a mathematical limit resulting from what is obtained by the expression:
Such a result is classified as an irrational number, since it cannot be represented in the form of a ratio. Its algebraic manipulation generally occurs by referring to it by its symbol, that is, the letter “e”. Or, in numerical applications it will depend on the precision you want to achieve, as in the example shown in the table below.
Going another way, we propose to investigate this mathematical entity through a geometric analysis.
Firstly, it is important to work with the hypothesis that “e”, Euler’s constant, is a number that resides in the interval between the numbers two and three, that is,
It is worth mentioning that the expression of the mathematical limit representing Euler’s constant is a potentiation operation, as seen in
to which we can apply Newton’s binomial formula to the numerator of the fractional expression. Then we can rewrite:
or, to simplify,
where
and therefore we can conclude that
Let’s analyze the “bₙ” portion. It should be noted that it can be interpreted as the sum of the terms of a non-regular geometric progression. Even so, we can determine the expression of the general ratio “q” between two consecutive terms of order “p” and “p + 1”. We then have,
whose development and simplification results in:
Obviously “qₚ” is less than 1, as it is enough to understand that “n - p” is less than “n”.
We can also, as a methodology, name an initial term “b₀” of this sequence:
With this term, we can now write:
or,
where
The algebraic aspect of the expression of “c” suggests that we can make use here of the results of resolving the dichotomy paradox, that is, writing:
and, for its ceiling value, we then have
Therefore, we can certainly write:
Bringing together the results obtained, we now have:
and, passing this inequality to the limit, one arrives, as a consequence, at the desired result, that is,
Another consequent result is that, for x > 0, we can write:
This can be observed in the Cartesian plane, as suggested by the diagram in the figure below, where the respective tangent segments appear purposely at x equal to 1, suggesting that there is also a geometric restriction at play, which we will investigate.
Sliding the value of n, starting from n equal to 1, reveals some important characteristics.
In the figure above “d” is a trigonometric circle (radius equal to 1), the length of FJ represents the value of the trigonometric tangent of the arc centered on B, this is the extreme point of the tangent segment of the corresponding curve of the function that at point A (x = 1). The length of the segment A’A corresponds to the theoretical value of the vertical coordinate of f(x) at x equal to 1.
In the figure below we have the same graph containing the trace of trigonometric tangents as the value of n is increased.
Based on the elements of the graphs, we highlight the following observations:
- point B tends to be congruent with point O, which is at the origin.
- The FJ segment tends to be congruent with the A’A segment.
- The segment FJ will never exceed the segment A’A, otherwise this would destroy the circle “d”, that is, the circle “d” will park its center, in any case, on the point O.
- The mathematical limit that defines Euler’s constant (“e”) has a geometric correspondence, i.e.
Such observations lead to the following conclusion:
Euler’s constant represents the value of the vertical coordinate in the Cartesian plane of the unit absissa point of the theoretical exponential curve, whose tangent segment at this point has as its extremity exactly the origin of the Cartesian axes, and which is necessarily equal to the value of the trigonometric tangent of the tilt angle.
