Derive Definition of Exponential Function (Power Series) using Taylor Series
The exponential function, denoted in the form and the shorthand of , can be defined using a Taylor Series expanded around . There are two important properties of the exponential function that are used. The first is that and the second is that the exponential function is the derivative of itself. The general form of the Taylor Series is given below.
Steps

Recall, the exponential function has two unique properties. 1) The value of the function at is . 2) The output of the function is equal to rate of change of the function. Or in other words the function is the same as its derivative.

Given below is the general form of the Taylor Series. The variable represent the point around which we are expanding. Since we know the value of the function and its derivative at , choosing is a natural choice.
Expanding the summation and substituting into the formula we get the infinite series below. Note, the direct substitution of gets translated to the notatin of to represent the first order derivative.

Next, start by finding the first, second, third and so on derivatives of the function. In the case of the exponential function, this is trivial. Note, for this example refers to .
Then, let’s calculate the value of the derivatives at . This is pretty straightfoward, they are all equal to .
Subsituting the values into the series we get the new expression.
Lastly, a little bookkeeping. Let’s move the polynomial expression into the numerator and evaluate the factorial expressions. Note, and the expential property property .

In conclustion, the Taylor Expansion of the Exponential function is given by power series below.