# What is the value of e?

##### 1 Answer
Apr 22, 2015

e = 1+ 1/1 + 1/(2*1) + 1/(3*2*1) + 1/(4*3*2*1) + 1/(5!)+ 1/(6!) + * * *

(For positive integer $n$, we define: n! = n(n-1)(n-2) * * * (3)(2)(1) and 0! = 1

$e$ is the coordinate on the $x$-axis where the area under $y = \frac{1}{x}$ and above the axis, from $1$ to $e$ is $1$

$e = {\lim}_{m \rightarrow \infty} {\left(1 + \frac{1}{m}\right)}^{m}$

$e \approx 2.71828$ it is an irrational number, so its decimal expansion neither terminates nor goes into a cycle.
(It is also transcendental which, among other things, means it cannot be written using finitely many algebraic operations
($\times , \div , + , - , \text{exponents and roots}$) and whole numbers.)