 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. In lecture 35, we're starting the last stretch of this lecture series. This will correspond to chapter 11 of Jane Stewart's Calculus textbook, where we begin talking about the ideas of sequences in series. This is all building up to the notion of a power series, which we'll end this lecture series with here. And so there's a lot that's going to go on in these last several lectures. And in some regard, it's going to feel very different from what we've been doing previously, but it's actually going to be trying my goal over the next couple lecture videos to convince you what we're doing is actually not fundamentally different than what we did before, but there is a slight change of the switch as we go forward. More sense as we get into the meat of things here. So in section 11.1, we're going to introduce the notion of a sequence, where a sequence is a type of function. A sequence is a function whose domain is taken over only positive integers. Sometimes that's fudged and we allow the number zero to be inside the domain, but we'll just take a domain of positive integers right now. And so take as an example the following sequence. We're going to take the function f of n given by the function 1 over n, but we only allow positive integers for our domain. So this function is defined at f of 1, f of 2, f of 3, f of 4, et cetera. Now, if we evaluate f of 1, we plug it into the function here. We put 1 in for the n, we get 1 over 1, which is a 1, and so we get this value here. So f of 1, we equal 1, and as 1 is the first number in the domain, we kind of think of 1 as the first number in the sequence. Well, what about f of 2? f of 2 means you're going to plug in 2 for your input variable. We get 1 over 2, which is 1 half, and so we get here. And so since 2 is the second number in the domain, 1 half as the second number in this sequence, right? f of 3, the computation here is the same. We're going to replace n with a third. So then the third number in this sequence would be 1 third. Continuing on here, f of 4, we would plug in a 4 as we evaluate this function. And hence, the fourth number in this sequence would be 1 fourth. And this sequence then can continue on and on and on. We get 1 over 1 fifth, 1 over 1 sixth, 1 over 1 seventh, 1 over 1 eighth. And then this continues on and on and on as far as we want to go. Now, some conventions we should be aware of as we work with sequences. Because sequences are functions, but in order to distinguish them from the real valued functions we'd seen before, whose domains consist of intervals of real numbers, there's some notational conventions we're going to modify for the situation. One of them we already saw is that when we talk about a sequence, we actually don't typically use the number x. We're actually going to typically use the symbol n here instead. n is meant to stand for a natural number, a natural number meant to say there. And the symbol x is typically here going to represent a real number. Real numbers are the numbers we've seen many times before. This include whole numbers, fractions, decimals, both terminating decimals and infinite decimal lengths, non-repeating decimal lengths, irrational, rational numbers and things like that. The natural numbers are we're only talking about whole numbers and we're restricting ourselves to positive integers and zero. Like I said, sometimes we do allow our sequences to start at zero. Although in this video, most of the future ones will be starting at positive one, right? So we're going to typically use the variable n instead of x in the context of sequences. So we know that we're only taking these positive integers as our input. Another thing we often do differently with sequences, instead of using this like f of n notation, we generally will use a subscript of some kind like we might say a sub n is equal to 1 over n. So because again, we like to think of this idea of a sequence as a list of numbers like we have here. We have this a1, this a2, this a3, etc. We like to think of the number as this list, this infinite list, mind you. We have this list of numbers that we're going to write out. And so this is a common convention of describing what we might say is the general form, the general form of our sequence. But these things are functions and as such, we can treat them like functions in many, many ways. For example, we can graph these things. Now when it comes to graphing a sequence, these points are isolated in the plane. You have the first point, 1 comma 1, the second point, 2 comma 1 half, the third point, 3 comma 1 third, the fourth point, 4 comma 1 fourth, etc. And then you're gonna see there's this big gap that sits between all of these points. There's like this gulf that sits between the first point, the second point, the second point, the third point, the third point, the fourth point, etc. And so what you see here in yellow is what's part of the sequence. Now you can see that there's this dashed green line here as well. So the yellow points, this represents the sequence a sub n equals 1 over n. The green dash you see is actually the real continuation. We're taking the function f of x to be 1 over x. That is, if we were to sort of connect the dots in a continuous smooth way, we could extend the sequence 1 over n into the real valued continuous function 1 over x. And so this is an important distinction here. In the past, we've been talking about continuous functions, such as f of x equals 1 over x. As we transition to sequences, we're now talking about what's referred to as a discrete function. A discrete function, sort of a difference between continuous. Discrete has these gaps that sit between the points. There's gaps between the data as opposed to a continuous function, which there aren't gonna be any gaps between them. Hence the idea of a continuous function forms a continuum of points. While these discrete points are somewhat isolated in space. And so chapter 11 is starting really the idea of discrete calculus. As we look at these discrete functions, aka sequences, we will be then transitioning in the next section and subsequent sections into talking about discrete integrals, which we will be calling series.