Taylor Series Revisited: An Introduction Related to Numerical Methods

In this segment we will revisit Taylor series, and just get you introduced to it. In a way we are reintroducing you to Taylor series in this segment. So, if you look at the general Taylor series that is given by this expression. And when students sometimes look at a Taylor series, they get intimidated by it. But if I put it in simple terms, what is Taylor series? Taylor series is simply that hey, give me the value of the function at a single point. Keep in mind that it's not asking you for hey give me the function, give me the value of the function at a single point. And give me its first derivative at that particular point, second derivative, third derivative, and what can happen then? I can give you the value of the function at any other point x plus h. Of course, there's a fine print in in this, and that fine print is that all these derivatives and the function has to exist and be continuous between x and x plus h. And that is the beauty of Taylor series. Not to be thought as a complicated formula, but as something which allows us to calculate the value of the function at different points.

What are the applications of Taylor series? One is to calculate transcendental functions. So, what I mean is for example your e to the power x you must have seen in your calculus series course, formula like this. What is this? This is a Taylor series which is given for e to the power x around the point x equal to 0. And it has been reduced to a polynomial and as we know that polynomials are very easy to calculate in a computer because it requires basic arithmetic operations of multiplication, addition, division, and subtraction so that's one reason why we talk about Taylor series. The second reason we talk about Taylor series is because it allows us to derive numerical methods. Not all of them, but some of them.

So, you'll find out that hey let's suppose if I want to find the second derivative of a function, an approximate formula for it, you can derive it from Taylor series. If you want to derive let's suppose a method called Newton Raphson method for solving nonlinear equations, you can derive that by using Taylor series. And we'll be doing all these things in the in the rest of the course. The third one is finding accuracy of numerical methods. So, what we mean by that is that if you are deriving or if you are finding a formula, let's suppose, for a numerical method, would you be able to say how accurate that formula is? And Taylor series allows you to do that to find out what the order of the accuracy of a numerical method is. And then the beauty of it is that there are some methods which you can use which will help you to use that knowledge not only to find the order of the accuracy of the numerical method, but also to come up with more efficient numerical methods.

So, you extrapolate based on the order of the accuracy you can extrapolate to come up with a better numerical method than what you have an example of that is, let's suppose, you are finding the second derivative of a function as I mentioned to you. Once you find the accuracy of that particular formula, you are able to use something called the Richardson's extrapolation formula to come up with the more accurate estimate of your derivatives of the function. And all that will be done in the course. And this is the end of this segment.