e – Sam Artigliere's Blog

There are a number of definitions of the exponential function, $e$ . However, there are 3 main ones:

$e=\displaystyle\sum_{k=0}^\infty \frac{1}{k!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots$
$e = \displaystyle\lim_{n\to\infty} = \left(1 + \frac{1}{n}\right)^n$
$e$ is the positive real number such that $\displaystyle\int_1^e \frac{1}{t}\,dt=1$

Here is a proof of #1:

We showed elsewhere on this site that the Taylor series representation of $e^x$ equals $e^x$ for all $x$ . This representation is as follows:

$e^x = \displaystyle\lim_{n\to\infty}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$

But $e=e^1$ . Thus, $x=1$ . The limit shown above then becomes:

$e^x = \displaystyle\lim_{n\to\infty}\frac{x^n}{n!}=1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\dots$

The following link provides an intuitive explanation then a formal proof of #2:

The following link provides a proof that the 3 definitions of $e$ listed above are equivalent:

To understand this proof, videos from series on combinatorics and the binomial theorem taken from Khan Academy may be helpful: