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!

$e_{x}=∑_{n=0}n!x_{n} =1+1x +2x_{2} +2×3x_{3} +2×3×4x_{4} +…$