MacLaurin series of Exponential function, e^x

The MacLaulin series (Taylor series at x=0) representation of a function f(x) is
Definition of MacLaurin series

The derivatives of the exponential function e^x and their values at x=0 are:

derivatives of e^x

Note that the derivative of e^x is also e^x and e^0 = 1. We substitute this value of f^{(n)}(0) in the above MacLaurin series:
MacLaurin series of e^x

We can also get the MacLaurin series of e^{ix} by replacing x to ix:

MacLaurin series of e^{ix}

e^{ix} is used in Euler's Equation.