 In this video, I want us to consider the series, which actually has a variable inside of it. Let's find the sum of the series. The series is this infinite sum here, where n ranges from zero to infinity of x to the n. So some things to point out here is that first of all, oftentimes our series will start at n equals one, but in this situation, we want it to start at zero. There's plenty of reasons why we might wanna include zero. It's sort of just a starting point. We have to specify it. So that's, you see at the bottom of the series there. But then what's also different about this one compared to previous examples, is that there'll be a variable inside of it. We're taking x to the n. What does this add up to b? Well, because this is a geometric series, where your constant ratio here is gonna be x, and your initial term a is gonna equal x to the zero, which will equal one. This is a geometric series right here, and therefore we can use the formula for a geometric series. We're gonna see here that the sum as n equals zero to infinity of x to the n, this will equal the first term, which let me solve as x to the zero. And then we see on the bottom one minus the constant ratio, which is x right here, we get one minus x in the bottom. This simplifies to be one over one minus x. And so what we see right here is that this geometric series, because of the variable present, this is often referred to as a power series. This geometric series is actually equal to the expression, the formula of one minus x here, where this is a formula for x, right? And so what we can see as long as the absolute value of x is strictly less than one, it turns out that this function, one over one minus x, is actually equal to this power series. And it turns out that this connection is gonna be quite, what's really important as we go further into this chapter. In fact, this type of connection, connecting a function to a series is really the main reason why we are gonna be studying series. Because as we've talked about, sequences are examples of discrete functions. A discrete function, that's what we mean by a sequence compared to continuous functions we saw in the past. A series is actually an example of what we might call a discrete integral. This will become much more clear in the future here, but series, that is the sequence of partial sums, the limit of the sequence of partial sums can be viewed of as like an integral, because after all, integrals, as we first learned about them in the continuous setting, these are limits of Riemann sums, where the Riemann sum in some respect is a sum of the original function. Well, that's what a series is. A series is also an infinite sum, where we add together the terms of the sequence. Now in a Riemann sum, we want our rectangles to get thinner and thinner and thinner so that delta x gets smaller and smaller and smaller. In the discrete setting, we can't just make the terms get closer and closer and closer because there's this gap of one that sits between all the terms. And so because of that, this series, we really don't wanna think of it as integrals, these are discrete integrals. And that's why we study series in calculus two. Calculus two is all about integration. And series are just a different type of integral from the continuous integrals we spent in the previous part of our course studying here. And so what we're gonna see here is that one can represent a continuous function using a discrete integral. And there's gonna be a lot of power when one makes that observation.