linguistic:translation
Power series definition

For complex z e z = 1 + z 1 ! + z 2 2 ! + z 3 3 ! + ⋯ = ∑ n = 0 ∞ z n n !. {\displaystyle e^{z}=1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}

Using the ratio test, it is possible to show that this power series has an infinite radius of convergence and so defines e for all complex z.