Calculus: Using the Limit Comparison Test

limit

09/16/2010

~ Romy

excellent job explaining!!! just excellent! the $2 i spent on this has probably been the most worthy investment ive ever made!

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Sequences and Series
The Limit Comparison Test
Using the Limit Comparison Test Page [1 of 3]
So the limit comparison test is a wonderful way of figuring out if a particular infinite series converges or diverges, if you can sort of see within it sort of the basic spirit of what's making it do whatever it does and then compare it to that spirit. In particular, if you have two infinite series and it turns out that the limit of the quotient of the individual terms of that series exists and is positive, then either both series converge or both series diverge. So if you know one of them, then you automatically know the other one as well. So this is a great technique to use whenever you're looking at infinite series and you see within it a familiar infinite series. This technique of limit comparison tests is really one that is worth considering.
So let's take a look at some examples and see some of these ideas in action. Let's look at the infinite series . Okay, that looks pretty complicate. It's not quite clear how to look at this. The terms themselves do go to zero, since if you use L'Hospital's rule once, you'll see that the terms actually shrink to zero. So the quickie test doesn't help us here, but do they shrink to zero fast enough, that's the question, for it converge or does it shrink down to zero sufficiently slowly, so that, in fact, this thing diverges? Who knows?
Well, let's see if we can compare it to something. Now, what can we compare it to? Well, I'm actually going to make a bad choice, just to show you what it looks like if you make a bad choice. A lot of times we'll make bad choices. That's how we learn the difference between a bad choice and a good choice. Well, let's try to compare it with . I happen to know that that converges, it's a p-series, where the p = 3, which is bigger than 1, so this converges. Let's compare and see what happens. If we compare, we need to take . Again, the order, which one you put on top and which one you put on the bottom, really does not matter, because, in fact, in the actual description of this particular test, you just take two infinite series, take their quotient, and if the limit is between zero and infinity, strictly between zero and infinity, then they either both converge or both diverge. It doesn't make any difference which one you know and which one is the mystery one. So you can put them any way you want. I'll just put them this way just for fun. When I invert and multiply, then the n^3 comes on top, migrates up north, and so what I see here is an .
Now, you can use L'Hospital's rule or just look at this and realize that, on top, this thing is growing like n^4 after I distribute, and, on the bottom, I'm growing like n^3. So a few successive repetitions of L'Hospital's rule, or just thinking about it, we see the top is growing faster than the bottom, so this limit actually is infinite. This limit is infinite, and so what does the test say? Well, the test actually just says absolutely nothing, because the hypothesis is that that limit L has to be finite and positive. And this is an example where, in fact, the limit is infinite. That does not mean anything diverges or converges, it means the test has failed. It's inconclusive and so therefore we say nothing.
Well, so maybe, in fact, this was a bad choice of our comparison series. And, indeed it was, because if you really pick up on this theme that I'm trying to invite us to think about together, this theme of finding the spirit of it, this is not the spirit of that. This is the spirit of something else, potentially, but not the spirit of this. The spirit of this would be to say, "How are things growing?" The top is just growing like n and the bottom is growing like n^3. Now, in the old example, I just dismissed the top completely and just looked at the bottom. But I can't dismiss the top, there's growth in the top and the growth is like n. So, in fact, this becomes , which you notice, by the way, is still a p-series, where the p > 1, so this still converges, but it's a different infinite series than the original one we tried to compare to. So let's now compare our unknown one with this one, thinking that this is really the true spirit of the original.
So let's take . So what does that equal? Well that equals the limit, as n goes to infinity, when you invert and multiply, I get the n^2 on top, so I get . And so this limit, I'll write out one little step here more, if you distribute, you see . You can apply L'Hopital's rule a few times or realize that, in fact, the growth is the n^3 on top and the n^3 on the bottom. So, in fact, in the limit, this is going to go just to the quotient of the coefficients, which is . So you can work this out and verify that this, in fact, is . , that's the limit, that' L, the limit of the quotient. That limit is positive and not infinite, and so therefore they either both converge or both diverge. I indeed found the true spirit of this infinite series. And since I know that converges by the p-series test, I know that this also must converge by the limit comparison test.
So the moral is this converges by the limit comparison test and comparing it with . , we saw, didn't work, but this one actually does work. And notice how, when I write my answer, I also give a little rational. The whole thing is my answer for that one.
Let's try another one together, . Okay, let's just cut right to it and see if we can figure out what the real spirit of this infinite series is. Well, let's see, the 1 I don't care about, actually. Even the 2 I don't care about, all I care about is the growth. There's no growth on top and the growth here is . So this growth spirit seems to me to be , which is the sum of . = n. And that's the harmonic series that diverges. And so if I was to actually compare this diverging series with this series and I see that basically they're comparable when I take the limit, then, in fact, since this diverges, that would diverge as well.
All right, let's take a look and see how well they do compare. So let's take the limit, so these turn into limit problems now, so we take the limit. And I'll put this term here, again, order does not matter, you can put either one on top. So I have . And so when I invert and multiply the n, it comes up on top, so what I see is . Now I've got to compute that limit.
Now you can try to use L'Hospital's rule here and I'll let you try it and see how well that goes. It's appropriate, allowed and see what happens. But it's going to be not great. Try it and see why. Instead, let me actually do a little trick here. That square root is the thing that plainly bothers everyone. Everyone is annoyed by the square root. That's not good. So how can we get rid of the square root? Well, we can't. So instead, let's try to put a square root on top. So, in fact, let's try to make a square root happen on top. Now, I'll have to have an equal sign here, so I can't change anything. I can't just put a square root there. So if I put a square root there, I've got to immediately take it away by writing it as an n^2. Notice that n^2 with a square root is just equal to n. In fact, that was exactly the official first version of the infinite series that we guessed to compare to. Only later in life did we realize it was . So, in fact, let's go back to that now.
And now why is this potentially a good idea? Well, because now, since I have the square root of top and square root of bottom, I could write this as just the square root of the whole thing. And the reason why that's good is because now what I can just do is take the limit of the inside and take whatever answer I get and take the square root. And what's the answer? Well, if you use L'Hospital's rule just on this thing right there, if I cover up the square root part like that, what you see is, well, you'll get , because the growth rate of top and bottom are the same, so I just look at the coefficients. I see under the square root, and so the answer is . So that's the limit.
That limit, by the way, you'll notice is positive and a finite number, and so therefore, since it's positive and finite, whatever one series does, the other series follows suit. So, in fact, we really did find the true spirit of this particular infinite series. The true spirit of it really is . So therefore, since this diverges, I know this diverges, and thus what I can report here is that this diverges by the limit comparison test, comparing it with .
Neat! So this is a great test, where you don't have to worry about inequalities anymore. You just find the essence, take a limit, and if the limit is positive and finite, then whatever one series does, the other one follows. I'll see you at the next lecture. Congratulations!