 Hello, and welcome today to a screencast about the convergence of a sequence. All right, so just a little refresher here. A sequence, s sub n, and it's in these little curly brackets to remind us that it's a sequence, converges to a number l if we can force values of s sub n beyond some point to be as close to l as we want. We write the limit as n approaches infinity of s sub n equals l. So all of those limit ideas that you learned back in calc one, we're going to be using again here in calc two. So let's say we can't find such a number l, then we say the sequence diverges. All right, fantastic. So number one, determine whether the sequence converges or diverges, and we're going to do this one by hand. The next one, we're going to use some technology to do it. So we've got the sequence 1 minus 2n over 5n plus 4. So our s sub n, we could say then is 1 minus 2n over 5n plus 4. And if we look at the limit as n approaches infinity of 1 minus 2n over 5n plus 4. Okay, so this is where, again, we've got to dig back into our calc one knowledge. So if I were to look at the numerator here, what's going to happen to the numerator as n gets really, really big? Well, this is also going to get really, really big. What happens to the denominator as n gets really, really big? It's also going to get really, really big. Does anybody remember what happens when we have infinity over infinity? We've got to use Lopitalz. Right, hopefully you remembered that. Okay, so this is going to equal, I'm going to write a little lh here so we remember we're using Lopitalz. I still have my limit as n approaches infinity. So remember Lopitalz says we're going to do the derivative of the numerator, and then we're going to do the derivative of the denominator separately. So we don't use any quotient rule or anything like that. So the derivative of my numerator is negative 2. The derivative of my denominator is 5. Guess what, my n's go away. So this limit is just going to be negative 2 fifths. Okay, is negative 2 fifths a number? Absolutely. So we say that this sequence converges. Okay, number one is done. And remember, you didn't have to use Lopitalz on this particular example. You could have divided each piece by n just to see what happens there. So there are lots of different ways to do it. I'm just trying to get you to remember some of the stuff you learned back in Calc 1. Okay, our next example, and I'm going to bring up just the tippy bottom here, says, forget it, I'll bring up the whole thing. Okay, determine whether the sequence converges or diverges. This time we're going to be using technology. Okay, and that's because our sequence is n factorial over 2 to the n. Yeah, there's no way we're going to be able to do this one by hand. I mean, I can kind of like ration it out or rationalize it out to myself, but yeah, doing this one by hand, we can't use Lopitalz. We can't really use any nice algebra. So what I've done here in Geogebra is I had Geogebra create the first, I guess it's probably eight terms of our sequence here. Eight or nine, I forget how many we got here. But anyway, we've got the first few terms of our sequence. And if you guys have never seen this factorial before. So let's say, for example, our s sub 5 is going to be 5 factorial over 2 to the 5th. Okay, so what that factorial means, and there's a button in your graphing calculator in your math menu. So that means we're going to do 5 times 4 times 3 times 2 times 1 all over 2 to the 5th, which is 32. Okay, so we end up getting, when you do all that out, that's, I believe, this value here of that 3.75. Okay, so just so you know how that factorial function works. So anyway, as you can see here, as we're looking at these terms, holy cow, wow, this one here got ginormous. And that factorial is going to grow really, really fast. Okay, so what I did in Geogebra is I went ahead and I plotted out our different points here. Just so again, you can kind of see the general shape, and you could definitely tell that this thing is going to be increasing and it's going to keep on increasing. Just to convince myself to make sure it didn't kind of level off or anything, I went ahead and graphed the function x factorial over 2 to the x, because these two functions are very much related. So remember, this function with the x's is going to be continuous. It's going to cover all those different values. Where our sequence is going to be discrete and it's only going to cover the values at our whole numbers, our integers. Okay, so that's the difference between the two of them. But you can always take a look at the continuous function to get an idea of what's happening with the discrete function. So anyway, hopefully you guys can see this function is going to grow crazy fast. So our denominator is growing pretty big too, right? It's an exponential function. But the problem is that numerator, that factorial, it's growing way, way faster. That's actually one of the fastest growing functions that you could come up with. So we're going to say here then that this thing diverges because there's no way we could find one number that this sequence is going to approach as our n values get even bigger and bigger. Alright, thank you for watching.