Calculus: Invert Series in Limit Comparison Test

This lesson has not been reviewed.

Please purchase the lesson to review.

Sequences and Series
The Limit Comparison Test
Inverting the Series in the Limit Comparison Test Page [1 of 4]
Let's continue in our journey through infinite series, and it seems like we have infinite many in the series to look at, and indeed we do, so let's get going. Let's take a look at the following infinite series, and remember, the only question at hand is does it converge or diverge? We've strayed away from the more interesting and more natural question: if it converges, what does it converge to? We just don't understand that question at all. So, in fact, we have to make do with this thing, it converges.
Summation of . So let's immediately notice that the terms themselves go to zero, so the quickie test is inconclusive. Can't integrate that function too easily, so the integral test might not be good. Now, can I compare it to something? Well, all the terms are positive. Always got to make sure that all the terms of positive, by the way, otherwise you can't use any comparison test. Comparison tests only hold when you have positive terms.
What can we possibly compare this thing to? Well, what's the spirit of this series? Well, downstairs here, the n^4 is a much more rapidly growing function than the . In fact, that really dominates and takes control. So for all thinking purposes this thing's not even there in my mind. That's the way I want to have your intuition begin to be reformed; that in fact, this thing, since it's so much slower growing than this, that this takes over, throw that away. So the spirit of this, in my mind, is just this. And that's a p-series where p in this case is 4, it's bigger than 1, so it converges.
So my thinking is I'm going to try a limit comparison test on this series with this known series to converge. If the limit is positive and finite, then whatever one does, the other one will do. In this case if this converges, that would force this to converge as well. That's the strategy. Let's see how this plays out.
What I'll do is, I'm going to take the limit, as n goes to infinity of . There's the mystery series. I divide it by the known series. Again, I just make a difference how I place this in. I'll write it this way, but you could have written it the other way, too. Either way is fine, doesn't make a difference as long as the limit is positive and finite.
So what's that? Well, I invert and multiply, and so I get the limit as n goes to infinity. The n^4 now migrates on the top, so I see this, n^4 + 2. And now you've got to take that limit. Now you could apply L'Hôpital's rule a whole bunch of times, and that's fine, and you can try that and see what you get. But you know what? I don't even want to bother with that.
I get to this point and say, "You know? This doesn't look that great to me." What would look better in my mind, is if these guys would have been inverted, if I would have switched the top and the bottom. So maybe this is an example where I'll actually elect not to put this on top and this on the bottom, but to switch them. Since it doesn't make a difference in the test, if it's more convenient to take the limit the other way, let's do it.
So instead, let's sort of hold, so put this into a holding pattern. We'll table it. That means it's still there, but maybe will never be picked up again. We'll just table that and try the other direction, because that limit seems too hard. It's computable, but it might be too hard. So always keep your eyes open for making life a little easier on ourselves. So , I'm going to put that on top, and then put the on the bottom. Again, doesn't make a difference where I put which as long as they compare.
Now what's that? That's the limit as n goes to infinity. If I invert this and multiply it comes on top. . So it looks exactly like what we previously had, except inverted. In fact, you may have just wanted to go right to this step and say, "You know what? I won't do that limit. I'll do the inverse of it and get to here." But I wanted to show you the steps that really all I'm doing is switching the roles of these guys to get to here.
Why is this so much better than this? It seems like it's the exact same problem. Well, it's not exactly the same problem, because now I can break that up into two fractions. I can break it up into the + . Notice that really is a legal move, because since I have the same bottom I can just add the tops and I get this. So I'm breaking up this fraction into these two fractions. I couldn't have done that trick here.
But now what happens? Well, the , they cancel, so I'm just left with one, and then what happens here? When I cancel this away, this is n^1/2 - let me write this little teeny piece out here so you can see that. This is 2 n^1/2 divided by - and that's n^4, and n^4 is just n^8/2. That's just n^4. And so what does that equal? That equals 2, and I can cancel. If I cancel I'm just left with n^7/2 on the bottom.
And what happens as n races off to infinity? Well, this thing goes to zero. So in fact, this piece right here just goes to zero, and so I immediately see the limit is one without doing any L'Hôpital's rule even. So swapping was a good idea. The limit is one.
Remember the rule, as long as the limit is positive and finite, then we win. So one is certainly positive and it's certainly finite, so that means that whatever one of the series does, the other series must do the same. This series is known to converge. Therefore, this must converge as well. So this converges by the limit comparison test, comparing it with the p-series . Neat. Look how quickly. And that's a complicated looking one, so it's amazing how we can do these things.
Let's try one last one together here. Summation n going from one to infinity of . There we go. Now we've got a little transmittal function action here. Looking really threatening. How would you proceed? Well, again, I'm just searching for the very soul, the very spirit of this thing. Let's see what we get.
What's the essence? Well, the real question is, if you're asking essence questions, is how is this thing growing like? So who's winning out in the race? Is the growing in a faster rate or slower rate than the ln n? Well, it turns out that the , in fact, n raised to any power, like n^.001, actually will always beat the ln^n. The ln^n is a really slow growing function. So n to any power - in this case we see n^1/2 - will always beat the natural log.
So when you let this thing race off really all the action's here, so that might be a good thing to compare to. This is a p-series where the p is 1/2. That's less than one, so this actually diverges. So we know this diverges, p-series. So if we compare to this and it compares that, in fact, the growth rate is the same, then we know this must diverge as well. The guess is this will diverge and let's see if we can actually make that guess into a reality using the limit comparison test.
So let's go. Let's now take the limit as n goes to infinity of what? Well, I'll take this term and I'll divide it by the term that we're looking at here, . So what does that look like? Well, that looks like the limit as n goes to infinity. If invert and multiply, the comes on top, and I have on the bottom + 2ln^n. And you know what? I get into the same sort of paradox I had before. This is an indeterminate form with a variety infinity over infinity, and I don't know what to do. I can use L'Hôpital's rule a few times. Might get stuck though. It would be so much nicer if these things would have been flip-flopped. So I now I realize that in terms of taking the limit, from the limit point of view, it might have been better to switch these two things, even though in terms of the actual mathematical point of view using a limit comparison test it doesn't matter.
Let's actually do that right now. So I'm trying to show you that you shouldn't be afraid to flop them if you think it makes a difference. So let's take two. If we take two, then I see the limit as n goes to infinity. First I'll put the known one, and I'll divide by that the unknown one. Let's see if this make a difference in taking the limit or not. Don't be afraid to experiment with these things and try other possibilities.
If I invert and multiply now I see . Now, I can break that up happily into two pieces, + . So what does that equal? Well, that equals just a one, so that's just a one, and now I've got to take this limit. So this limit is the limit as n approaches infinity of .
Now, maybe you're thinking to yourself, "Well, the is supposed to win out, and if the wins out, then this thing should go to zero." Well, that's actually correct, but you can now verify that for yourself. Back when I told you how the grows faster than the natural log, if you didn't believe me, good for you. You shouldn't believe anything I say. In fact, you shouldn't believe anything that anyone says unless you understand it yourself. Otherwise, be in doubt.
Now let's verify that with a simple application of L'Hôpital's rule. So if I use L'Hôpital's rule the derivative of the top is just 2 divided by n, and the derivative of the bottom was the derivative of the . That's going to be , and so if you now clean this up, if you invert and multiply this 2comes up on top, so that's going to be 4divided by n, which equals the limit as n goes to infinity of 4 divided by the , because this is just the times , and so I get a cancellation. That limit now you can see is zero.
So indeed, what I said before was actually correct and now you see it. This grows so much faster than the natural log, that this pulls everything down to zero. This is going to infinity faster than this. The whole thing is zero. This term is zero, so this limit equals just 1 + 0, which is one. That's finite. The positive. So therefore, whatever one of the infinite series does, the other one does. We compared it to , which we know diverges. So therefore, I conclude so this series will diverge as well by limit comparison test with summation .
It's really a terrific test when you look at something that looks complicated, but you really see the essence of it. If you see the essence of it and you can verify that it really is the essence, meaning the limit of the quotient is not zero, but finite, then you actually know that the thing will converge or diverge.
Congratulations on really conquering this limit comparison test. It takes a while, but just give it time, let it sink in, and you can do it.