= Taylor expansion definition of the exponential function
{c}
The <Taylor series> expansion is the most direct definition of the expontial as it obviously satisfies the <exponential function differential equation>:
* the first constant term dies
* each other term gets converted to the one before
* because we have <infinite> many terms, we get what we started with!
$$
e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + \frac{x}{1} + \frac{x^2}{2} + \frac{x^3}{2 \times 3} + \frac{x^4}{2 \times 3 \times 4} + \ldots
$$