The exponential function 𝑒𝑥𝑝(𝑥) can be expanded into a power series:

• $exp(x):={\sum }_{k=0}^{\infty }\frac{x^k}{k!}=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...$

Proof

See: Taylor Series Expansion of Exponential Function

Other

• Can be used to derive Euler’s Formula