Definition:Euler's Number

As the Limit of a Sequence
The sequence $$\left \langle {x_n} \right \rangle$$ defined as $$x_n = \left({1 + \frac 1 n}\right)^n$$ converges to a limit.

That limit is called Euler's Number and is denoted $$e$$.

Its value is approximately $$2.718281828 \ldots$$

As the Limit of a Series
The series $$\sum_{n=0}^\infty \frac 1 {n!}$$ also converges to the same limit.

As the Base of the Natural Logarithm
The number $$e$$ can also be defined as the number satisfied by $$\ln e = 1$$.

Equivalence of Definitions
See Exponential as the Limit of a Sequence for how $$\lim_{n \to \infty} \left({1 + \frac 1 n}\right)^n = e$$ follows from the definition of $$e$$ as the number satisfied by $$\ln e = 1$$.

See Euler's Number: Limit of Sequence implies Limit of Series for how $$e = \sum_{n=0}^\infty \frac 1 {n!}$$ follows from $$\lim_{n \to \infty} \left({1 + \frac 1 n}\right)^n = e$$.

Now suppose $$e$$ is defined as $$e = \sum_{n=0}^\infty \frac 1 {n!}$$.

Let us consider the series $$f \left({x}\right) = \sum_{n=0}^\infty \frac x {n!}$$.

From Series of Power over Factorial Converges, this is convergent for all $$x$$.

We differentiate $$f \left({x}\right)$$ WRT $$x$$ term by term (justified by Power Series Differentiable on Interval of Convergence), and get:

$$ $$ $$ $$

Thus we have $$D_x \left({f \left({x}\right)}\right) = f \left({x}\right)$$ and thus from Differential of Exponential Function it follows that $$f \left({x}\right) = e^x$$.

From Derivative of an Inverse Function we get that $$D_x \left({f^{-1} \left({x}\right)}\right) = \frac 1 {^{-1} \left({x}\right)}$$.

Hence from Basic Properties of Natural Logarithm it follows that $$f^{-1} \left({x}\right) = \ln x$$.

It follows that $$e$$ can be defined as that number such that $$\ln e = 1$$.

Hence all the definitions of $$e$$ as given here are equivalent.

Comment
This is the most famous irrational constant in mathematics after $$\pi$$, and equally far-reaching in scope and usefulness.

The proof that it is irrational is straightforward.