e

I think I had a breakthrough with how mathematicians can find out why and which irrational numbers arise in a multitude of crazy situations. One of them is with the definition of Euler's number, e. Here is the limit definition of e:

lim(x→∞) [1+1/x]^x

The function of e^x can be expressed as the following:

1 + x + x²/2! + x³/3! + x⁴/4! + .......

Which implies that "e" or e¹ is the following:

1 + 1 + 1/2! + 1/3! + 1/4! + ......

Here is how you can find the relationship between the limit and the infinite series.

Rewrite lim(x→∞) [1+1/x]^x as lim(x→∞) (x+1)^x/x^x. Now expand the numerator to a representative infinite series.


(x^x + xx^x-1 + x(x-1)x^x-2/2! + x(x-1)(x-2)x^x-3/3! + ......)/x^x

If we divide out x^x, we are left with:

1 + 1 + x(x-1)/(x²2!) + x(x-1)(x-2)/(x³3!) + ........

Then, by taking the limit as x approaches infinity, the subtractions off of x in the numberator become insignificant and cancel out the corresponding x in the denominator. The result... which is absolutely beautiful, I believe... like, seriously... I'm in tears...

1 + 1 + 1/2! + 1/3! + 1/4! .......

e = ∑(1/x!), [x,1,∞]