How is it the complex exponential converges for any value of $z$ in the complex plane? $e^{z} = 1 + \frac{z}{1!} + \frac{z^2}{2!} \cdots\cdots$
How is it the "sum of exponents" rule holds for complex exponential, that is $e^{w}e^{z} = e^{w+z} $?
...using only the definition of $e^z$?