Taylor Series Skill Instructions

Invented by James Gregory, the Taylor Series is an incredibly powerful tool in Calculus.

James Gregory

Image Credit: Wikipedia



Formula

Taylor Series Formula

Explanation

As the sigma shows, the Taylor series is an infinite sum of terms. Any function - sine, cosine, logarithm, et cetera, can be expanded into a Taylor Series. Then, you just plug a certain x in to get out the value of the function at that x. The more terms you retain, the more precise it is. Now, to explain the formula: a is any real or complex number - you choose. x is just x - you don't have to substitute anything there. ƒ is the function that you are trying to expand into a Taylor Series. n is a number that increases for each term in the Taylor Series - so, there is a first term for n = 0, second term for n = 1, third term n = 2, et cetera.



Example

Let's do an example with sin(x):

First eight terms of the Taylor Expansion of Sine

Here, we are using the formula for the first eight terms of the Taylor Expansion. a = 0 for the sake of simplicity. When a = 0, it is a special case of the Taylor Series which is known as a MacLaurin Series. It can be seen how, for each term, n increases by one - so, for the first term, n = 0; for the second, n = 1; et cetera. This expansion can go on infinitely, but we are only retaining eight terms.

When we simplify this, we get:

First eight terms of the Taylor Expansion of Sine, simplified

Notice that the terms where n is even got lost, as they all equal to 0. This is not the case for all functions.

The more terms we retain, the more precise we are. As we go farther and farther from a (which, in this case, is 0), the graph of the Taylor Polynomial deviates farther and farther from the graph of our function. Therefore, adding more terms helps us compute the function even as we significantly increase the distance from a. This can be seen by the graph of our current Taylor Expansion in comparison with the original graph of sine:

Taylor Series Graph

The red wave is sin(x), and the blue curve is our eight-term Taylor Polynomial. It can be seen how, close to 0, the polynomial overlaps with the sine wave. However, if we want to calculate the sine at, for example, x = -4, the overlap would be insufficient, and we would need to add more terms to be precise.

Here is a video demonstration of the Taylor Expansion:

Video made by me

Click here to see applications of the Taylor Series and further resources.