The defining characteristic of the exponential function is that d/dx e^x = e^x. The series construction makes this really obvious, because each element of the s

The defining characteristic of the exponential function is that d/dx e^x = e^x. The series construction makes this really obvious, because each element of the series is the first derivative of the next element, ad infinitum. And it wasn't as if this series was somehow luckily discovered, it was constructed from the defining characteristic. Visual Complex Analysis, by Tristan Needham, covers this lucidly.