e^{i x}=\sum _{n=0}^{\infty } \frac{i^nx^n}{n!}=1+i x-\frac{x^2}{2!}-\frac{i x^3}{3!}+\frac{x^4}{4!}+\frac{i x^5}{5!}-\frac{x^6}{6!}-\frac{i x^7}{7!}+\frac{x^8}{8!}+\frac{i x^9}{9!}-\frac{x^{10}}{10!}-\frac{i x^{11}}{11!}+\text{...}