 Welcome back to our lecture series, Math 1220, Calculus II, for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misalign. This is our first video for Lecture 46. And so I do want to warn you that the videos in this lecture are going to be a little bit longer than usual. There's not as many lectures or many videos for 46 because of that. That's because we want to talk about the very important topic of Taylor's inequality. So we learned last time about Taylor's theorem, which tells us that if a function has a power series representation, then that power series will necessarily be given by the formula you see right here. So f of x, so well, I should say that what we see is that the function, the power series would be the sum of cn times x minus a to the n, such as the power series centered at a. And then the formula would be given as this. The coefficient sequence would look like the derivative of f evaluated at a divided by n factorial. That's what the formula for a power series representation has to look like. But then the question I want to explore today is how do we know that a function is actually equal to its power series representation? Now, the reason why this if is such a big deal is we'll actually talk about in the last video for this lecture here. The last video actually will give us an example of a function which is not equal to its Maclaurin series for any number except at the center of the Maclaurin series, which is zero, of course. So the question about if a function is equal to its Maclaurin series or any Taylor series is a big deal because the answer is sometimes no. But we're going to talk about some situations that can help us know when a function is equal to its Maclaurin series or some other Taylor series. And we are actually going to do this using what's called Taylor's inequality. So Taylor's inequality is the following. This is actually, it's going to look like an error bound formula that we've seen in the past, which actually it is an error bound formula. And in section 11.11 of this series, so not lecture 46, 47, excuse me, but in lecture 48, we'll actually use Taylor's inequality as error bounds. So we'll talk about that a little bit later. But for this lecture today, we want to use Taylor's inequality to show that certain functions are equal to their Maclaurin series. So what Taylor's inequality tells us the following. As first, we have to find an upper bound for the n plus first derivative of f. So if the absolute value of the n plus first derivative of f is less than or equal to m for all x's that are no more than d units away from the center. So the distance between x and the center is less than or equal to d right here, right? Then the remainder of the Taylor series is bounded above and satisfies Taylor's inequality right here. The remainder r in of x is going to be less than m over n plus one factorial times x minus a to the n plus first, while x again is sufficiently close to the center there. So let me give you a little bit of more explanation what's going on here. So like we've done with series so many times, if we were to approximate a series, we often use a partial sum to do that. Well, what if we wanted to do the same thing for a power series? We wanted to approximate a power series. Well, in that situation, we're going to use the notation t sub n of x. And if this is a partial sum, we would take the sum where k ranges from 0 to n of f, or the k-th derivative of f, evaluated at a divided by k factorial, that's Taylor's coefficient there, times it by x minus a to the k. This is the formula for the Taylor series. The only difference is that instead of going towards infinity, we only go off towards n. We only go toward n there. And as such, this sum is a finite sum. And this right here, this partial sum of a Taylor series is commonly referred to as a Taylor polynomial. A finite power series is a polynomial, much in the same way as I often describe a power series as an infinite polynomial. And so this is in fact the nth degree Taylor polynomial. And so what we see here is that this remainder rn of x, you'll notice how it's rn of x, right? The remainder depends not just on how many terms in the sum, but also depends on x itself. And so therefore, rn of x, we're going to define this to be the absolute value of f of x minus tn of x, okay? So what we know for a fact here is that the tn of x is the Taylor polynomials, they're going to converge towards the Taylor series, which we'll call that capital T right here. The Taylor polynomials will converge to the Taylor series. Now, if the series is equal to the function f of x, that means the Taylor polynomials will converge towards the function itself. So we want to look at the error between the function and the Taylor polynomial. How close can this Taylor polynomial get to the function? The remainder rn of x, this measures the error between the function and the Taylor polynomials. If the Taylor polynomials converge towards the function, then this would converge towards zero. And so that's what this rn is trying to measure right here. And therefore, we want to show that this thing goes to zero and one can use Taylor's inequality to accomplish that. So that gives you an explanation of Taylor's inequality. What we're going to see in the next two videos is I'm going to show you how you can use Taylor's inequality to prove that e to the x is equal to its Maclaurin series, and then we'll also show that sine of x is equal to its Maclaurin series. And so check those out in the next two videos. They'll be a little bit longer. That's okay. Take a look at those on the links. You shall hopefully see right now.