 Hello! Welcome to this quick recap of section 8.1 on sequences. Chapter 8 is the start of a different part of this class. Things will look quite different from what we've done so far. Sequences are our first example of what we call discrete information. So let's take a look at a definition for it. A sequence is just a name for a list of terms in a specified order. By terms, we usually mean numbers. For example, here's a sequence 1, 4, 9, 16, 25, and so on. The dots here indicate that this sequence continues on in the same way forever. Sequences are often infinitely long. Here's another sequence. This one's called the Fibonacci sequence. Each number here is the sum of the two before it. A sequence can represent all sorts of information. This one could represent how much money is in a bank account on the first day of every month. But sequences don't have to have any sort of pattern. This one shows a sequence can have any values at all, and it could continue on forever with no particular pattern. As long as those numbers come in a specified order, here 9 is the first one, negative 2 is the second, pi is the fourth, and so on, then this is a sequence. Sequences represent what we call discrete information, that is, individual numbers that come in a definite order. A sequence in its terms can be named, and understanding these names is key to being able to work with the sequence. If we put the term s of n in curly braces, we refer to the entire sequence that is all numbers in the order that they go in. We can also name the individual terms of the sequence. The first one is named s sub 1, the second one is named s sub 2, and so on. If we want to talk about a generic term, especially if we know a pattern for it, we can talk about s sub n, the generic term or general term in a sequence. It's important to know the difference between the curly braces notation and the no curly braces notation. With curly braces, we're talking about the entire sequence and all numbers in it in order. Without curly braces, we're talking about a single term, just one number in the sequence, and we can do any sort of algebraic operations we would like to for that. Here's an example of this. If we put n squared in curly braces, the sequence n squared would be 1, 4, 9, and so on. The general term would be called n squared. If we wanted to, we could write s sub n equals n squared. Notice how this time I don't have curly braces because I'm telling you the value of an individual term. This value lets me calculate the value of any term I'm interested in. For example, s sub 999 would be equal to 999 squared. A sequence can be infinite or finite. A finite sequence is just one that ends eventually, while an infinite one continues forever. The sequence above here for n squared is an infinite one, and most of the sequences we're interested in will be infinite. Just like functions, we have an idea of a limit and convergence for sequences. A sequence s sub n is said to converge to a number l if we can force the values of s sub n beyond a certain point to be as close to the limit l as we want. We'll write this notation, which is pronounced the limit as n goes to infinity of s sub n equals l to represent when a sequence converges to a limit. If a sequence doesn't converge, then we say it diverges. The sequence s sub n below, starting one, one-half, one-quarter, and so on, converges because we can see that its terms are eventually going to get as close as we want to zero, and they'll stay within that tolerance, getting closer and closer to zero. So we would write that the limit as n goes to infinity of this term s sub n is equal to zero. On the other hand, this sequence t sub n goes to infinity as n gets larger and larger, and so we would say this sequence diverges. But there's many ways that a sequence can diverge. For example, the sequence u sub n simply bounces back and forth between one and zero, one and zero. Even though this doesn't go to infinity, it still diverges because the sequence never settles down and gets as close as we want to any one number. It doesn't get close and stay close to any number, but it bounces back and forth. Now that we've seen these definitions, let's take a look at some of them in action.