Sequences
Primary tabs
SAMPLE LESSON
Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:
A sequence is a function whose domain (recall domain is the x-axis) is the set of positive integers and whose range (recall range is the y-axis) is the real numbers. There are several things we want to note about sequences.
First, we want to add some detail to the domain. We define it as the set of positive integers, but other places may define it as the set of natural numbers. Recall, that the natural numbers can include $0$ or not include it. And is zero a positive integer or a negative? The point is that we can define the domain to be <Sign in to see all the formulas> or <Sign in to see all the formulas>. It doesn't matter which we use you just need to be aware that different textbooks may define the domain differently. Some may say natural numbers where there definition of natural numbers may or may not include $0$. Don't be overly concerned by this because it is just a notational difference.
Second, how we denote a sequence typically comes in two different types. Since a sequence is a function you sometimes see it denoted by $S(N)$ where $S$ is for sequence and $N$ is for natural numbers. However, you will more than likely you will see it as $S_N$ to emphasize the discreet nature of the domain. In other words, you may see <Sign in to see all the formulas> as possible notations for a sequence that an author will be talking about.
The other way to denote a sequence is with a set-like notation <Sign in to see all the formulas>. (Note that from our discussion about the domain we could just as easily said <Sign in to see all the formulas>.) So let's do a concrete example. The sequence <Sign in to see all the formulas> is also denoted by <Sign in to see all the formulas>. We can tell from the context that we have defined the domain to be <Sign in to see all the formulas>.
Third, when referring to the $S_N$ we say that it is the Nth term of the sequence. For example, $S_4$ will the the fourth term in the sequence.
Fourth, a sequence can be bounded. Recall the definition of a bounded function. Since a sequence is a function it can also be bounded. In other words, a sequence is bounded if there is a number $B$ such that <Sign in to see all the formulas>.
Finally, we have talked a lot about the domain and whether it includes $0$ or not. The reason this doesn't matter is because the most important aspect of a sequence is whether it converges. We will talk about that in the next lesson, but what is important to us about a sequence is not the terms at the beginning, but the behavior of the numbers as $N$ in $S_N$ becomes larger and larger. Does $S_N$ get closer and closer to some number or not? If it does then we say the sequence converges. Understanding whether a sequence converges or not is the most important thing to understand about a sequence and that is why having $0$ in the domain (or not) will not matter to us at all.
