A Taylor Series is a tool in mathematics to define a function as an infinite power series. A Taylor series approximates the function around a point and uses the first, second and so on derivatives of the function to solve for higher-order polynomials. A Taylor series is the general form of a Maclaurin Series, which always expands around the point, zero.

The general form of a Taylor Series of a function expanded around the point is given below. As the number of terms approaches infinity so does the accuracy of the approximation on the condition that the series converges. The general form of the Taylor Series of a function is shown below.

This is also written using the summation symbol (capital sigma).

Shown below are some functions that are defined using a Taylor Series approximation.

The exponential function can be defined using a Taylor series as shown in this example.

The sine function can be defined using a Taylor series as shown in this example.

The cosine function can be defined using a Taylor series as shown in this example.