In this article, we will consider the Taylor Series. We will try to get an intuition of what it is used for, as well as how to calculate it. We will also introduce the concept of the remainder.
Intuition Behind The Taylor Series
The Taylor Series is a series that is used to create an estimate of what a function looks like. So, what exactly does this mean? The theory is that if a point is chosen, i.e., an x-coordinate and a y-coordinate, then it’s possible to estimate what the function looks like in the area around the point. How is this done? It is done by taking the derivatives of the function and adding them all together. Then, when you have an infinite number of derivatives added together — then you will end up with a single finite sum.
Now, let us try to see an example of how this would look. Afterward, we can look at how we will calculate the Taylor Series. Imagine that we want to find the Taylor Series, i.e., the approximation of x² around the point x=1. Then the first three terms would look like so:
We can then follow how the function changes with each term added until, in the end, it completely resembles the original function.
