Calculus: Monotonic and Bounded Sequences

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Sequences and Series
Monotonic and Bounded Sequences
Monotonic and ounded Sequences Page [1 of 2]
When you think of a sequence, of this collection of numbers just one after the other and so forth, like a list of numbers, I guess, in some sense, you can sort of think about what they look like by just plotting the points, like this, for example. You know, you have n = 1, n = 2, n = 3, n = 4. And you can just plot what the first term in the sequence is and just put a dot at that number, and the second term, you can put a dot at this number, and wherever they are. So you actually sort of plot a sequence if you want. If you think of the plots like this, when we can actually talk about some sort of interesting - or maybe not interesting - but maybe useful ideas.
For example, there's one idea of monotonicity. And so I say that a sequence is monotonically increasing in the values constantly go up. So as you go off - as the index grows - the terms themselves grow. For example, an example would be n^2 - so 1^2, 2^2, 3^2, 4^2. Those terms are actually getting larger and larger and larger. I could write it this way, so a[1] is smaller than the next one, a[2], which in turn, is smaller than the next one, a[3], and so forth. So you can see that the values are actually growing as I go out. I'd say this is a monotonically increasing sequence. Now the question is, can you figure out what I mean by a monotonically decreasing sequence? I think you can.
It would be just this. The terms start off pretty high and then get smaller and smaller and smaller and smaller. So a sequence is monotonically decreasing if a 1 is the biggest one, and then the next biggest one would be a 2. And by the way, whether you put an equal sign here or not - it depends upon your own personal tastes. I won't put them in. And a[2] is actually greater than a[3]. And so they're shrinking as you go down the list. For example, like , , , , , those numbers are getting smaller and smaller and smaller. That's an example of a monotonically decreasing sequence. So that's the idea of monotonically increasing or decreasing.
Now, what I'll think of is if you think of this plot and you look at a plot like this, for example, where it's sort of just both increasing and decreasing, I'll say that a sequence is bounded from above if I look at this plot and I imagine this plot going on forever, if I could find some value for which the terms never escape beyond. So a sequence is bounded from above if they all live below a particular value. For example, the sequence, 1, , , , that sequence is actually bounded from above. It's bounded from above by the number 1, say, or 2, because there's no term in that sequence, , that exceeds 2, or even that exceeds 1. So that's an example of a sequence that's bounded from above. There's some feeling of which all the terms live below that ceiling. I say that a sequence is bounded from below if I can find a floor for which all the terms in the sequence live above that floor. For example, let's look at the example. That's a good example - . That's , , , and so forth. Notice that all of those terms, for example, are greater than zero. So zero is a floor under which none of those terms in that particular sequence will dip below. So the sequence is an example of a sequence that's bounded from above and also from below. So it lives in some sort of band.
The sequence n^2 - so 1^2, 2^2, 3^2, 4^2 - that's not bounded from above, because that's going off to infinity. But it is bounded from below because you never get any values say less than 1. So in fact, that sequence - the n^2 sequence - is bounded from below but not bounded from above. So in fact, you can think about bounded from below, bounded from above, and so forth.
So let me just take a look at an example. How about this sequence right here, the sequence . Now, first, what's going on here? Well, when you think about this, as n goes through its values, so n = 1, n = 2, n = 3, what happens to this number? When n = 1, this is -1^1,[ ]which is -1. When n = 2, this is -1^2, which is -1^2[ ]which is 1. When I have n = 3, this is -1^3 is actually -1. In fact, what does this thing actually do? This basically is just a switch that goes between -1 and 1. It just oscillates back and forth, -1, 1, -1, 1, depending on whether n is even, in which case this is 1, or odd, in which case this is -1. So this toggles back and forth between -1 and 1.
And so in fact, what I see here is that this starts off at -1. So the first term, for example, would be -1 + . So that's the first term, which is just a zero. The next term would be 1 + . The next term would be -1 + , and so forth. That's the particular sequence that I'm describing here.
Now in fact, this sequence is an example of a sequence that is - first of all, is this increasing - monotonically increasing or monotonically decreasing? Well, actually it's neither. This does wiggle back and forth a little bit. It just is not solely climbing or solely falling. This is an example of a sequence that's bumping up and down. And I guess you could sort of see that. I start with zero and then I go up to , but then I go down to . So you can see right there. Here I'm going up but then from here, I go down. So it certainly is not always going up or always going down. It wiggles back and forth, in fact.
However, even though it's wiggling, this is an example of a bounded function. This function is bounded from both above and below. In fact, it's bounded from above by 1. This thing will never, ever, ever, ever get bigger than 1. And it's bounded from below by -5, because this will never, ever, ever, ever be smaller than -5. Now I'm not saying those are the world's best bounds, but certainly, as long as I can find some bound for which they live in between, then I can say that this function is bounded.
And by the way, if I say that a sequence is bounded, and I don't say bounded from above or below, it means that you bound it from both above and from below. That's what it means to be a bounded sequence. You're bounded from above and below.
So I think that's about all I wanted to tell you about bounded sequences and the idea of monotonically increasing sequences and so forth and so I guess that's all I have to say.
Oh wait, wait, but one last thing. I almost forgot. I'm so absent-minded. Suppose that you have - I'm sorry to bother you one more second - suppose that you have a sequence and it's bounded and it's monotonically increasing, how would that have to look? Now I don't mean to trouble you with this. But suppose that I'm going to put a sequence down here. So I'm going to actually put the dots down, but they have to be climbing, and yet there's a ceiling. So I can never, ever, ever, ever, ever in life get above that line. I'm not saying that I actually have to get to that line, but that's going to be some sort of upper bound. I can't get above it.
Now I'm going to start to put the points down. Here's a challenge for you right now. Put down points and always go up and never go above that line, what has to happen? Well, what has to happen is, no matter how hard you try, the thing will have to taper off somewhere. It might not be at that line, but it will have to taper off somewhere. And in particular, when something tapers off, that means that when I go and look through my binoculars, I actually see that that sequence has a limit. It's got to taper off. It can't just zoom off to infinity, because it's bounded by here. It can't wiggle, because it's only increasing. So since it can't zoom off to infinity, and since it can't wiggle and serpentine, what must happen is, it's got to go up and it's got to somehow level off somewhere. It might level off at that line. That's a possibility. Or it might level off somewhere before. But the bottom line is it would have to level off, which means that the limit of the sequence exists. You can find the limit.
So whenever you have a sequence that's increasing and bounded, then it definitely has a limit. Similarly, if you have a sequence that's bounded from below and decreasing, it too must have a limit. So in fact, we can see that if we have things that are increasing but bounded from above, there will be a limit. Now what that limit is, that might be hard to define. But anyway, I didn't mean to bother you with that, but I just wanted to tell you about that. So I'll see you at the next lecture. Bye.