Calculus: Introduction to Limit Comparison Test

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Sequences and Series
The Limit Comparison Test
Introduction to the Limit Comparison Test Page [1 of 2]
In the comparison test, what one does is if you're trying to figure out if a given infinite series converges or diverges, what you hope to find is a familiar series, call it , and if that familiar series converges and dominates the unknown series, then the unknown series must converge. And if that familiar series diverges and is dominated by the unknown series, then the unknown series must also diverge. So that's great, but then you have this sort of domination requirement. You have to get these inequalities to work out just right. And how do you actually find the familiar series? You find the familiar series by trying to find the true spirit of the infinite series at hand. And it turns out that there's a technique for testing whether infinite series converge or diverge, which actually doesn't require the domination thing. So if we want to throw away the inequalities, then we look at what happens in the true spirit of the thing, and that true spirit can be thought of, in terms of a test, as the limit comparison test. So let me describe to you now the limit comparison test.
Suppose we're given an infinite series and we want to figure out whether that infinite series converges or diverges. And, as always so for in our discussions, I'm going to assume that these are positive terms. And now pretend I have a familiar one, or I have some other one, and again, positively termed values. Now, to see whether these two things have the same spirit - what does that mean? Well, what I'm asking for is basically are they growing at the same rate? Are the individual terms, the a[n] term and the b[n] term, are they growing at basically the same rate? If they're growing at the same rate, let's just be sort of artsy here for second, if they're growing at the same rate, then whatever one infinite series does, the other one should follow suite, because they're basically growing at the same rate. It's like a horse race, where they're sort of all going in a big clump. So if one converges, the other one must converge, if one diverges, the other one should diverge as well. It doesn't make a difference which one it is. If this converges, that should converge, if that diverges, that should diverge, and vice versa. So basically, if these guys are growing at the same rate, then in my mind I'm thinking about that they have sort of the same spirit. That's what I'm looking for. How do I see if they're growing at the same rate? Well, what I'll do is I'll compare them. I'll weigh one against the other and then see what happens as I let the parameter n go to infinity.
So here is the limit comparison test. The limit comparison test says let's take the limit, let's see what happens as you head off to infinity, so . So I'm going to compare these two values, so I'm taking the n-th term here, the n-th term here, and take their quotient to see how far off they are from one another, and then let that go to infinity.
Now, suppose that that limit exists, and I'll call the answer, the limit, the number L. Then if that value L exists and it's not zero, so its number is positive, but not infinite, positive and a real number, then, in fact, they have the same spirit, which means they either both converge or they both diverge. Then either they both converge - they, I mean the series. You've heard of we the jury? This is they the series. Oh, I'm so good! They the series both converge or the other possibility is the series both diverge. So this is the limit comparison test, which means basically all I'm after is figuring out what the essence of one of these things is. And if you can find an infinite series where you know the answer, where you know that it converges, and then when you take this quotient and that limit is positive, it's very important that it's positive and not infinite, then if you know one of them converges, the other one must converge as well. If you know that one of them diverges, then the other one diverges as well. Whatever one does, the other one will follow, because, in some sense, they have the same spirit, they have the same growth rate. So they both either go to zero fast enough, or they don't go to zero fast enough, but they both do it together.
Okay, so this is a great test. There's no need for inequalities anymore.
So, let's take a look at an example, and I'll start off with sort of some modest examples. So here's one, this is . I want to figure out if it converges or diverges. Well, look, really the 5 is inconsequential, and so what I see here is that this essentially looks like . I know is a p-series, where p = 1, or I could think of it as the harmonic series. In either case, we saw, using the integral test, that that diverges. So now, if I compute this limit, so let's compare with this, as n approaches infinity of one of the terms of one of the series divided by one of the terms of the other series - and notice, by the way, it does not make a difference which one I put on top or which one I put on the bottom. You can put it anyway you want, it doesn't matter. What is that limit? Well, it's limit as n goes to infinity I invert and multiply. The n comes on top, the 5n is on the bottom, and so this is just the number after I cancel away. So the limit is . So the limit L = . And notice that does, in fact, satisfy the condition that it's positive and not infinite. So it's a finite number that's positive. So, in that case, I know that either both series converge or both series diverge. But look, I know this diverges, so therefore they both must diverge. So I'm using the limit comparison test to immediately say that since diverges, summation diverges by the limit comparison test. And I always like to write in a little reason. So when I think of the answer to a question of this sort, I not only say converges, but I try to say why, I'm sorry diverges by application of the limit comparison test and comparing it with .
So let's try another example, . Okay, well, this plus 1 thing, who cares? So really it seems to me that I should compare to , which is a p-series, where p = 2. That's bigger than 1, so we know that that will converge. So let's try to compare with . See how I'm looking for just the essence of the thing? I throw away the sort of insignificant terms and cut right to the chase, strip everything away and get a simpler series, where I know the answer. So let's write that down, known to converge. And so now let's do the limit comparison test.
So I take limit as n goes to infinity - and again, it doesn't make a difference what order you do this in. So you could put either this term on top and then divide it by that term or vice versa. It doesn't make a difference. It's your own choice or whatever is convenient. And sometimes in examples it will make a difference in terms of just taking the limit. One limit will be easy, whereas the reciprocal might be a little bit more challenging. If I invert and multiply, I get the n^2 on top, and then I have the n^2 + 1 on the bottom. That's an indeterminate form of the flavor , as I let n go to infinity, so I use L'Hospital's rule. And if I use L'Hospital's rule at one application, I see , because I just take the derivative of top and bottom. The n's drop out and I'm left with just 1, and so that limit equals 1. 1 is the limit and remember the limit comparison test says that as long as the limit is not infinite and it's greater than 0, then I know that whatever one does, the other one will do the same. And so since I know this converges, that forces this to converge as well. So I'd say that this converges by the limit comparison test with .
So, in looking for the essence, you really actually can figure out whether things converge or diverge through this technique of the limit comparison test. I'll see you at the next lecture.