 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 Misseldine. This is the first video in lecture 43 in our series, and this represents sort of the beginning of the end for our series here as we introduce the notion of power series today. So in the previous several videos, we've been talking about series, but a power series is essentially a series for which we insert a variable into the mix here. To be more precise, a power series is a series of the form. We're going to take the sum as n ranges from 0 to infinity of Cn times x to the n here, where Cn is some coefficient sequence, that is the sequence of numbers. x here is a variable, and we're taking powers of n. So in a more expanded form, this would look like C0 plus C1x plus C2x squared plus C3x cubed plus C4x to the fourth plus C5x to the fifth, et cetera, et cetera, et cetera. I should mention that in this discussion of power series, we are using the convention that 0 to the 0 is equal to the number 1. You'll notice, for example, that we omitted the number x to the 0, because that's typically a 1. There is one exception, right? When you take 0 to the 0, that's technically an indeterminate form, but when it comes to power series, we're always going to accept that number just to be 0 to the 0 is just 1 for convenience here. And so this is what we mean by a power series. Now a power series is essentially just an infinitely long polynomial. And we've actually seen as an example the previous one, previously we've seen the following example take the sum where n equals 0 to infinity of x to the n. Well, this right here would actually be a geometric series. You get 1 plus x plus x squared plus x cubed plus x to the fourth, et cetera, et cetera, et cetera. So we want to think of this like an infinitely long polynomial, in which case the coefficients of that polynomial are the terms in this sequence c sub n, and that's what we like to call out the coefficient sequence for that reason. And so yeah, we can visualize this geometric series as this infinite polynomial, a so-called power series. And the reason why they get the name power series is because we're taking powers of x. So we have 0x, the first x, the square, x cubed, et cetera, et cetera, et cetera. And we can also generalize this notion of power series a little bit. We could take the series where n ranges from 0 to infinity. Now, one thing you'll notice that's also true in this context that in the previous videos, when we talked about series, we nearly always started our series at 1. But in this context of power series, we nearly always want to start our series at 0. And this is because our functions are allowed some type of constant term. And so that will be took in place with n equals 0. So our series will start at n equals 0. So we take the sum of n equals 0 to infinity. And then take your coefficient sequence cn and times it by x minus a to the n. And so this essentially represents like a horizontal shift by a units from the expression we saw before. No big deal with that thing right there. And so if we wrote this in expanded form, we'd get c0. We would get c1 x minus a. We get c2 x minus a squared. We get c3 x minus a cubed, et cetera, et cetera, et cetera. And this is referred to as a power series. It's still a power series, but if we say it's centered at a, centered at a. Now the most common type of power series we're going to talk about is one of this form for which we would say that this power series is centered at 0, a equals 0. And so centering your power series at 0 is going to be one of the most important types of power series, one of the simplest types of power series. But a power series is an infinite sum, but it's really an infinite polynomial. It's a series with the variable x. We have powers of x in play here. And it turns out that the, our whole reason we want to talk about series in the first place really was to build this notion of power series. That I've said many times in this lecture series that sequences are these discrete functions, series are these discrete integrals. But why all of a sudden, why all of a sudden that we spent more than half the semester talking about continuous functions. Calculus one, we talked exclusively about continuous functions. Why all of a sudden switch to discrete? Well, it turns out it's quite self-serving here. Our whole reason we introduce series was so that we can get to power series. Because power series, although it's an infinite sum, this variable x could potentially take on multiple values, multiple real values. And so a power series itself is a continuous function we built using this discrete calculus we've developed. So the whole reason we wanted to introduce series is so that we can ultimately get to power series and then use power series as a tool to better study the continuous differential functions we've been talking about before. And we'll see some more about this connection between power series and regular continuous functions in forthcoming videos.