The Euler’s equation 𝑒^𝑖𝑥 can be expanded into a power series:

• $e^{ix}=\frac{x^0}{0!}+\frac{i{x}^1}{1!}-\frac{x^2}{2!}-\frac{i{x}^3}{3!}+\frac{x^4}{4!}+\frac{i{x}^5}{5!}-\frac{x^6}{6!}...$

Proof

See: Taylor Series Expansion of Euler’s Equation