Calculus: The Limit of a Sequence

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Sequences and Series
Sequences
The Limit of a Sequence Page [1 of 2]
A lot of times we're thinking about functions, so you plug in any number and then another number comes out and you can graph it and so forth and look at these things. What I want to think about now is what happens if we just look at certain numbers - certain sequences of numbers - so a pattern of numbers that we can sort of look at. And I'm going to think of a collection of numbers that go on forever and ever. I want to talk about an idea that's referred to infinite sequences.
So a sequence is literally just a collection of numbers that you list, one after the other, one after the other, and so forth. So, for example, a sequence that we're very familiar with the sequence of counting numbers: 1, 2, 3, 4, 5, 6 and so on. And the way I could write that actually out - I want to talk about the collection of all of them in that order - is to actually write it like this. Let me show you a way of writing the natural numbers. I could write it with a little brace here or a curly bracket. And then {1, 2, 3,...}. Now it goes on forever, so note that with "...". And that's the collection, of the sequence, of the natural numbers, 1, 2, 3, 4 and they go in succession. But there are a lot of other sequences. In fact you can obviously make up as many sequences as you have time to waste. How about this one? {1, 4, 9, 16...}. In fact, on the SATs you used to see these questions and had to figure out what's the next term in the sequence. Those are the kind of questions that I still have some nightmares about. In this case, if you look at these numbers and look for a pattern, you can see that basically all I'm doing is taking the next natural number, but squaring it. This is 1^2, this is 2^2, this is 3^2, this is 4^2. The next number, a good guess would be, 25, and so on.
So we could take a look at sequences, and in fact, you can write this in a very, very shorthandy kind of way. Suppose you wanted to actually take this with you and put it in your pocket. It's sort of big. It goes on forever. So one way is to sort of abstract this idea and to write this as a symbol here, to denote all of these things. And the technique, or the convention rather, is just to write a[n], to denote the n^th term in a sequence. So for example, a sequence might now look like this: a[1] - that means a[1] is the first term in the sequence. a[2 ]is the second term a[3] is the third term and so forth. And then a[n] would be n^th term. That's sort of a general looking term. But it keeps going and it goes on forever.
So in fact, this sequence can be expressed with the following formula: a[n] = n. What does that mean exactly? Well, a[1] - wherever I see an n, I put 1 - would be 1. a[2] would equal, well, whatever n is, which in this case is 2. So I see 2. a[3] would equal what I see if put in 3 here for n, and I see 3 and so forth. So in fact, this sequence could be expressed, if I wanted to, as just n - so a shorthand way of writing that. Or I could write it as a[n], where a[n] = n.
To try to get us sort of familiar with this idea of using this notation, this subscript tells me where I am in this list. You can think of a sequence as an infinitely long list. This sequence I could write this way and give it a name. I could call it something else. I could call it b[n], where - what do you think the n^th term is? How do I get the term that's in the first spot? I take the number 1 and I square it. In the second spot, I do the same thing. So, in fact, here the recipe that would generate this entire sequence would just be n^2. So if I just said, "I'm thinking of a sequence and its general term is n^2," you know it all. If I start with 1, I have 1^2, 2^2, 3^2, 4^2. The 16^th term in the sequence would be 16^2 and so on. So if I wrote b[31], what is that? Well, b is just the squaring thing, and so I square 31. So the 31^st person in this list would be 31^2, and so on. And you could make more elaborate examples, or course.
So these are all examples of sequences. Now, the question is, what, by the way, happens as you let this sequence drift off and go off to the horizon? Well, if you take a pair of binoculars, you can see and say, "What's that sequence doing as the n gets bigger and bigger and bigger?" Now, in this example here, what happens to these numbers? Well, they plainly get bigger and bigger and bigger. So in fact, these numbers are just approaching infinity, as you go off. In that case, we talk about the limit of the sequence. And in this case, the limit of the sequence would actually equal infinity, or we might say the limit doesn't exist, because it's not a particular number. In that case, we would say this sequence actually diverges. Because as you go further and further off, you're not heading toward any number, you're just getting bigger and bigger and bigger.
Similarly, this sequence - the sequence of perfect squares - that's also a sequence that diverges, because, in fact, as you can see, as I go out, I'm getting larger and larger and larger - so again, an example of a sequence that diverges.
Let's take a look at an example that is different from those two. Let's take a look at this example. Actually, I'll start to use the new notation and I'll see if we can figure out - here's a sequence, a[n], where the general term is equal to . So what's this sequence equal to? Well, this sequence is equal to - well what's a[1]. Well a[1], I just plug in 1 for n. So I see , which is 1. a[2] would be what I get when I plug in a 2 for n, so I see . So I'd see a half. a[3] would be . So basically, in the fourth spot, I see . In the fifth spot, I see , and so on. In the general spot, I see . But it keeps going.
So here's a sequence - these are the sequence of reciprocals, if you want to think of it that way. They're just the reciprocals of the good, old-fashioned natural numbers. And what's now the limit of this? Now if you look down here and take a look at this sequence, you see 1 and and and and . They're all getting smaller and smaller and smaller. They're always, of course, a little bit above zero. And in the limit, if n gets really, really big, what happens to this whole thing? It actually approaches zero. So this sequence actually has a limit. The limit of this sequence would equal zero. One way of writing that would be to say the limit as n goes off to the horizon, which means n goes to infinity, of a[n], because that's the name I gave it, that would equal zero. And what that means is, if you put in what a[n] equals, which in this case is , that means as n gets bigger and bigger and bigger, without bounds, the reciprocal gets smaller and smaller and smaller and approaches zero. In this case, if the limit exists, we say that this particular sequence would converge. So if, in fact, you're heading toward a target as you drift off to infinity, if you actually see something that you're approaching, then we say that sequence, in fact, converges. And if you're not approaching anything in particular - you're bouncing around - or if you're going off to infinity, then we say that the sequence diverges.
So one last little fun example for us to chew on here - I'm going to do this in red and I'll tell you why in a second. How about this sequence? {1, -1, 1, -1, 1, -1}, and that pattern continues forever. Now, if you look at this, you see the pattern. It just oscillates between 1 and -1. And so, in fact, as you drift off here in the sequence, you're not going to be getting really, really, really big or really, really, really small. Right? So when you look out with binoculars, in fact, it's very easy to see what's going on. You're -1 and then 1 and then -1 and then 1 and you go back and forth and you go back and forth. In fact, you just stay at -1 or you stay at 1 and we go back and forth.
Does this sequence have a limit? Well, are we approaching one particular number? Are we heading towards one unique target as I zoom off the horizon? The answer is no, because I'm wiggling between 1 and -1, 1, -1, 1, -1, and I wiggle forever. And since I'm doing infinitely many wiggling and I'm not honing in on one particular number, in fact, this particular sequence diverges, and not because the terms are getting really, really big, but because there's a lot of wiggling. There's too much wiggling.
So, for a sequence to converge, I have to have the terms come together and approach one particular number. And if the terms actually drift off to infinity - get arbitrarily large - drift off to negative infinity - get really, really tiny, with negative billion and negative trillion and so forth - or if they wiggle back and forth, then that sequence would diverge and wouldn't have a limit.
So that gives you a sense as to what sequences are. There's the collection of numbers that I can list in a particular order. And the question now is, what do those sequence of numbers approach? And so we're going to try to bring in some ideas of how to figure out what the limit of a sequence is - in particular, what happens when you take an infinite sequence and look and see what happens on the horizon. I'll see you there.