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Taylor expansion definition of the exponential function

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} + \dots

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Exponential map (Lie theory)