Taylor's inequality for the remainder of a series

This theorem looks elaborate, but it’s nothing more than a tool to find the remainder of a series. For example, oftentimes we’re asked to find the nth-degree Taylor polynomial that represents a function f(x). The sum of the terms after the nth term that aren’t included in the Taylor polynomial is the remainder. We can use Taylor’s inequality to find that remainder and say whether or not the nth-degree polynomial is a good approximation of the function’s actual value.

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How to build Taylor polynomials

Taylor series let us find a series representation for any function. In order to create a Taylor series representation for a function, we’ll need ‘a’, the value about which the function is defined and ‘n’, the degree to which we want to evaluate the function. Both of these are usually given in the problem. With a value for ‘a’ and ‘n’, we can build the chart below.

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