A series expansion of a function about an argument x0 is an expression of the form

A + B(x - x0) + C(x - x0)2 + D(x - x0)3 + ...

that can go on forever. These are amazingly useful.

The condition that exp x is its own derivative gives relationships between the coefficients, A, B, C,... that allow you to deduce what they all are quickly, given A.