CHAPTER 01.07: TAYLOR SERIES REVISITED: Introduction to Taylor Series
In this segment, I'm going to revisit Taylor series. So keep in mind that we are only revisiting Taylor series, this is not going to be a complete treatment of Taylor series, because this is, again, from your calculus series classes, sequence classes which you have taken. So if you want more information, you can go to your calculus 3 book and read about Taylor series, but I'm going to just recap the information which you have already learned from your Taylor series.
Now, I'm sure that you must have seen a series like this one, e to the power x is 1 plus x, plus x squared by factorial 2, plus x cubed by factorial 3. And this is one of the famous examples of a Taylor series, and in fact it's also called a Maclaurin series. Maclaurin series is a special case of a Taylor series. And this is an expansion of e to the power x, and you can find out that, in this case, when you are writing e to the power x, that it's written in terms of a series, which involves some simple addition, division, subtraction, and multiplications. So it doesn't involve any kind of use of transcendental, or exponential, or trigonometric functions, it just involves, the calculation of e to the power x simply involves the basic arithmetic operations which are available in the computer of multiplication, addition, division, and subtraction. So that's one of the reasons you might think that, hey, that's why Maclaurin series, or this Taylor series issue is important. But there are other reasons, also, why Taylor series is important, and you will be able to figure those out as you see separate segments of the videos which have been written about how Taylor series comes into the picture when you are talking about numerical methods. Taylor series has applications for numerical methods, whether you are solving a nonlinear equation, whether you are solving an ordinary differential equation, whether you are trying to find error in an integral, and so on and so forth, the application of Taylor series is endless, and that's why revisiting and knowing exactly what Taylor series is all about is extremely important.
Some other Maclaurin series which you might have seen are for sin of x is x, minus x cubed by factorial 3, plus x to the power 5 divided by 5 factorial, and so on and so forth. Another one which you might have seen is cos of x, cos of x is 1, minus x squared by factorial 2, plus x to the power 4 by factorial 4, and so on and so forth. So you have many, many examples of a Maclaurin series and Taylor series for different . . . different types of functions when you are talking about Taylor series. But this is not telling me what Taylor series is, itself. So let's go ahead and talk about what Taylor's Theorem is, in itself. It is given as follows. It says the value of the function at x plus h is given by the value of the function at x, plus the first derivative of the function at x times h, plus the second derivative of the function at x divided by factorial 2 times h squared, plus the third derivative of the function divided by factorial 3 times h cubed, so you get the idea, that the Taylor series is actually infinite series, where you have all these functions and its derivatives, which are showing up in the formula itself. Now, if you are going to explain Taylor series in very plain terms, what that means is that, give me the value of the function at a particular point, x. Give me the value of the function at a particular point, x, give me the value of the derivative of the function at that particular point, give me the value of the second derivative of the function at that particular point, give me the value of the third derivative of the function at that particular point, and all the other derivatives, which I have not shown here, then, if you want to find out the value of the point . . . of the function at some other point, which is h away from the point where you are giving me all this information, then I can give you that function. That's the beauty of the Taylor series. You've got to understand that Taylor series is not requiring you to give you the function itself, or the expression for the function. It is only asking you for the value of the function at a particular point, the value of the derivative at a particular point, the value of the second derivative at a particular point, the value of the third derivative at a particular point, and it's promising you that it can give you the value of the function at any other point. Now, this seems to be a miracle formula. Of course, it comes with a fine print, the fine print is that the value of the function and all its derivatives, so the fine print is that all derivatives . . . all derivatives have to be continuous and exist in x in this interval. So that's very important to know, that, although Taylor series is promising you something, that, hey, give me the value of the function at a particular point, give me the value of the derivative of the function at that particular point, second derivative, third, fourth, and fifth, and all the derivatives, and I can promise you that I'll give you the value of the function at some other point. This is the fine print, that all the derivatives, the function and all its derivatives have to be continuous and exist in the interval x to x plus h. If that's not the case, then Taylor's Theorem cannot be used. But keep in mind that it only has to be true in that particular domain. It doesn't have to be . . . outside the domain can do anything, but between the point where you are to the point where you want to go, you don't have to be . . . that's the only domain you have to be concerned about.
Another common myth which you find out about Taylor series is the fact that, hey, h has to be small. h does not need to be small, for the time the derivatives are all continuous and exist between x to x plus h, you can use this Taylor series, which has an infinite number of terms, going from x to x plus h, where h can be any number, it can be 1,000,000, it can be 2,000,000, it does not have to be small. The only reason why we talk about small numbers for h is because if h is small then we don't have to choose as many terms of the Taylor series, because if h is, let's suppose, 0.1, the contribution here will be 0.1, the contribution here will be 0.01, the contribution here will be 0.001. So that contribution starts decreasing from the terms ahead, and hence, in order to be able to get the value of the function accurately, you will have to use only a few terms. But there is no restriction on what the value of h is, and that's a common myth which many students have, that h has to be small, h does not need to be small, h can be anything you want to. And that's the end of this segment.