Informal discussion of the Euler's constant

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I'm going to resort to the concept of the integral without the formalism. If you derive the natural log the result is the inverse of [math]\displaystyle{ x }[/math]. The other direction, if you calculate the area under the function's curve given by the inverse of [math]\displaystyle{ x }[/math] you get the natural log. The reason for this specific function [math]\displaystyle{ f(x) = 1/x }[/math] is that deriving logs in any base that isn't [math]\displaystyle{ e }[/math] yields [math]\displaystyle{ 1/xb }[/math], where [math]\displaystyle{ b \neq 1 }[/math] and is some log. With the natural log we have that [math]\displaystyle{ \ln(e) = 1 \iff e^1 = e }[/math], which implies that if we integrate [math]\displaystyle{ 1/x }[/math] from 1 to [math]\displaystyle{ e }[/math] the result is 1. Why from 1? Because [math]\displaystyle{ e^0 = 1 \iff \ln(1) = 0 }[/math]. For now it suffices to say that the concept of the integral that we first learn is a sum of rectangles under the function's curve. The base of the rectangle is [math]\displaystyle{ x_2 - x_1 }[/math] and the height is [math]\displaystyle{ f(x_2) }[/math] itself. There is a matter about error in this computation regarding a small area between a rectangle and the function, but I'm disregarding the Riemann integral's formalism for now.

You can try to calculate area under [math]\displaystyle{ 1/x }[/math] with a calculator with brute force. Try a rectangle where the base is [math]\displaystyle{ 1.2 - 1.1 }[/math] and the height is [math]\displaystyle{ f(1.2) }[/math]. Sum the area of this rectangle with the next, base [math]\displaystyle{ 1.3 - 1.2 }[/math] and the height is [math]\displaystyle{ f(1.3) }[/math]. Repeat until you reach a number under 1 but close to it.