 In a previous video, we introduced the notion of power series, which are series that involve some variable x and then we take powers of x inside the series. Now, because power series are themselves series, the convergence of a power series is something we have to worry about. And the convergence of a power series is actually going to depend on the choice of x. There will be some choices of x that work and some that don't. Like, for example, if we take the geometric series, x to the n, where n goes from 0 to infinity here in expanded form, this looks like 1 plus x plus x squared plus x cubed, etc. Well, given that this is a geometric series, this is a geometric series, and in this geometric series, our constant ratio r is x itself. In order to be convergent, we need the ratio to be small, right? The absolute value of x would have to be less than 1. And so if we take, for example, x to equal 1 half, then the series would be convergent at that value. But on the other hand, if we take x to equal, say, for example 1, the series would be divergent in that situation. So the convergence of the series depends entirely on the choice of x, right? But admittedly, while x equals 1 might be caused divergence from this series, x equals 1 actually could be convergent for another series like this one right here we're going to see will be convergent at x equals 1. So, I mean, it depends on x, but in fact, we really could say that it depends entirely on the coefficient sequence, right? So this geometric series has a constant coefficient sequence of 1. It's always the number 1. But we could take, like, the factorial sequence. We could take the reciprocal of factorial sequence. And it turns out the convergence is going to depend on this coefficient sequence. So we want to figure this out. And so as given a power series with a specific coefficient sequence, we're going to ask ourselves what values of x make it converge? What's the domain of a power series? The domain, as we've talked about in the past, the domain of a function is every choice of x that makes the function evaluate into a real number. Well, for a power series to satisfy that condition, we need it to be convergent. And so really when we ask for what values of x is the given power series convergent, we're really asking for the domain of this power series. And so let's look at some examples. Let's start with example a right here. Let's take the sum where it equals 0 to infinity of the sequence in factorial x to the n. Now our good friend when it comes to finding the domain of a power series is going to be the ratio test. I mean, you could use the root test if you prefer, but the ratio test generally speaking will be a very effective tool to use in this context. And this is because since you have powers of x, the ratio will help you simplify these powers of x. The root test also works out very well too. And so it has a lot to do with the coefficient sequence where generally I think the ratio test will be a little bit simpler to do here. So with the ratio test, we have to look at the ratio of consecutive terms. We take an plus one over an. And so in this situation, we have to take the coefficient sequence and the powers of x. So we're going to get on top n plus one factorial times x to the n plus one. This will all sit above n factorial times x to the n. And this term we want to simplify and then take the limit as n goes to infinity. Now utilizing some factorizations on the top, we can factor n plus one factorial as n plus one times n factorial. The advantage here is that the n factorial on top cancels the n factorial on the bottom. And by similar reasoning, we're going to break up this x to the n plus one as x to the n times x, like so. And then we see that the x to the n cancels on top and bottom. And so in simplified form, this thing is going to look like n plus one times the absolute value of x. So notice what you'll notice what I did here, right? The absolute value of n plus one. I actually removed the absolute value just took in plus one as in has to range from zero to infinity in plus one will always be positive. The absolute value is just being redundant that situation. I did retain the absolute value around the x because in the moment, I don't know what x could be. It could be positive. It could be negative. It could be zero. It could be any real number potentially. So the absolute value here is going to be necessary for our calculation. Now when it comes to the ratio test, we have to take the limit as n goes to infinity of this ratio a n plus one over a n. We've now simplified that expression and we see that we're taking the limit as n goes to infinity of n plus one times the absolute value of x. In which case in this situation, our limit as n goes to infinity, notice that the absolute value of x has no effect on, it's not affected by the n. We can take it out as this constant, the absolute value of x times the limit as n goes to infinity here of n plus one. In which case this is going to turn into the absolute value of x times infinity, which most of the time this is going to be infinity, right? This will be infinity exactly when x doesn't equal zero. The issue is that when x equals zero, you get this indeterminate form zero times infinity in that situation, right? And it turns out that what happens here is that if x were to equal zero in this situation, so if this was zero, then n times n plus one times zero would always equal zero. And so as n goes to infinity, this thing always equals zero. And so in the end, you're going to end up with this equals zero in that situation. So what's the domain of the function in that situation? We see that for part a, the domain is going to turn out to just be the number zero. That when x equals zero, if you plug in zero to this thing, zero times n factorial will be zero. And if you take a sum of infinity and leave me zeros, you're going to get zero. This thing will be convergent at zero because this series will actually go off towards zero itself. Now, one thing you should notice is that for this power series, zero is actually the center of the power series. And you're going to see that the center of the power series is always part of its domain. But it turns out that's the only thing we get here. All right, so that's a first example. Looking at another example, we're going to see some things very different in this situation. And I guess I should mention that before we leave here, why is it significant that this thing goes off to zero? When you're using the ratio test, if this limit is greater than one, you are divergent. When it's less than one, it's convergent. And so normally this is going to be infinity, which is definitely bigger than one. The only way you can get less than one is when the limit goes to zero. And that happens only when x equals zero. A little bit of clarification there. A similar example we can see here, if we take the sum when n goes from zero to infinity of one over n factorial x to the n. We look at the limit of the ratios this time. It'll look very similar, a n plus one over a n. But as the n factorial is now in the denominator, this will look like x to the n plus one over n plus one factorial times n factorial over x to the n. Now the simplification you're going to see here is going to be strikingly similar. The x to the n's will cancel. The n factorial will cancel. And we're left with just the absolute value of x over n plus one. And now this time around, as n goes to infinity, you're going to see much different thing. Well again, the absolute value of x here, which we use at constant, this is going to go towards the absolute value of x over infinity. Which in this situation, as x can only be a finite number, if x is positive or negative, putting infinity in the bottom is going to squash everything. This is going to go off towards zero. But also in this situation, if x were equal to zero, you're going to get zero over infinity. There's no conflict there. Those both want to be zero. This thing is always going to equal zero. And in particular, zero is always less than one. So we actually get a very interesting observation here. In this situation, the ratio test says that this thing will be convergent independent of what x turns out to be. That this power series will be always, always, always convergent. There is no number which we can insert for x for which this thing won't be convergent. That's quite impressive here. And so in terms of domain, the domain of this function would be negative infinity to infinity. We would get all real numbers. And you see that's a bitter contrast, a very bitter contrast to a stark contrast to, we only got the center before. This power series right here is likewise centered at zero, but it actually gets all real numbers impressive. Looking at one third example here. Let's take the series where n equals zero to infinity as we add up x minus three to the n over n. And so again, using the ratio test, we're going to look at the limit of the sequence a n plus one over a n. This would look like we're going to take x minus three to the n plus one over n plus one. And we're going to times that by the reciprocal of n over x minus three, like so. And some cancelage that's going to happen in x minus three to the n will cancel to the x minus three to the n. That's in the denominator. I forgot to write the n there. We got it in there at the last moment when no one was looking. And therefore we can rewrite this as n over n plus one times the absolute value of x. You're going to notice that when you do these calculations trying to find the domain of a power series using the ratio test, you're always going to end up with this absolute value of x. You should expect that. Now as you take the limit here as n goes to infinity, this expression right here as it's balanced, it's going to go off towards one. You get one times the absolute value of x. And of course the absolute value of x, when is that thing less than one? This is actually somewhat of a question. The previous two examples, it didn't depend on the choice of x whatsoever, but this one does. It does depend on x, right? And I guess, whoops, I made a slight little mistake here. Let me go back here and fix this. We don't want the absolute value of x. We're actually supposed to get the absolute value of x minus three. Sorry about that, everyone. Make the correction. x minus three. And then we're going to get the same thing going on right here. We should be getting the absolute value of x minus three. We have to solve this inequality right here. And it's not too bad to do that. If you have the absolute value of x minus three is less than one, that implies that x minus three is less than one, but greater than negative one. And then adding three to both sides, we end up with x is less than, well, if we add three, you're going to get four on the right. And if you add three, you're going to get two on the left. And so this right here gives us the domain of what's going on here. Our power series here, which is a function, it'll be convergent from two to four. And we see we get something a little bit different here. But actually, now that you mentioned it, well, what happened? The ratio test only applies when you're less than one or greater than one. When you're actually equal to one, you're inconclusive. So it turns out we're not quite done here. We know that will be convergent between two and four, but what happens at two and four itself? Well, if we actually were to plug these into the function, right? If we take, for example, x equals four, then plugging four into the function, you're going to get the series n equals zero to infinity. You're going to get four minus three, which is one, one to the n over n. This gives you the series one over n as n goes from zero to infinity. I guess there's a typo here. We can't actually start this series at zero because it would be undefined. Again, no, for convergence, the starting value doesn't really matter here, which is why it doesn't significantly change much. But to be proper, we should start this series at one, not at zero. But anyways, you notice when you plug in x equals four, you end up with the harmonic series. And the harmonic series is divergent. And so it turns out that x can't actually equal four for this series to be convergent. So we actually get every number up until four, but we don't include four itself. But we see something different happen when we look at x equals two. Because when x equals two, you're going to plug this into the series n equals one to infinity. You're going to end up with two minus three, which is actually a negative one. You get negative one to the n over n. This is now the alternating harmonic series. And as it's an alternating series, the convergence is a lot easier to accomplish. By the alternate series test, this thing is actually a convergent series. And so that means we actually do converge at two. And so we kind of see this curious behavior that this function's domain will be two to four, where two is included and four is omitted. We are convergent from between all numbers, all numbers between two and four, which includes two but not four itself. And these three examples are kind of curious and actually provide for us a general expectation of what happens with power series. So it turns out that for a given power series, you take the sum, an infinite sum of cn times x minus a to the n. So this is a power series, just a generic power series centered at the number a. It turns out that the domain of that power series will be one of three possibilities, the first possibilities. The series will only converge at its center, x equals a, in which case the domain of the series will just be a single number a. On the other hand, though, you could write this as an interval. This is the interval from all numbers between a and a. I mean, it's kind of silly, but that is a possibility. And so that's that was like the first example we saw. Another possibility is that our series, looking at the second case, converges for all real numbers. In which case in that situation, our series would converge on the interval negative infinity to infinity, like we saw with our second example as well. And as a third possibility, it could be that there's some positive number r. There's a positive number r, so that whenever the distance between x from the center is less than r, the series will converge. And when the distance between x and the center is greater than r, then it diverges. In which case the domain will be one of four possibilities. You get all numbers between a minus r and a plus r, not including the endpoints. You have all numbers between a minus r and a plus r, including the endpoints. And you have these half and half versions where you have all numbers between a minus r and a plus r. You have the left end point, but not the right end point, or you have the right end point, not the left end point. The example we saw previously actually fell under this case right here. You'll notice that in all three situations, in all of these cases, the domain of the power series was an interval. And therefore the domain of a power series is commonly referred to as the interval of convergence. Because a power series will always converge on an interval. It could be a finite interval with either zero length. It could be an infinite one with infinite length, or something, one of these finite ones right here. And so the length of this interval of convergence we refer to as the diameter of convergence. If you take half of that length, you call the radius of convergence. The radius convergence is particularly important in this last case, where the radius in that case is going to be r. Because you can go r to the left, or r to the right of the center, and you're guaranteed convergence as long as you don't go more than r. At exactly r, things can get a little bit fishy. It turns out our first case also has a radius convergence. In that case, the radius convergence is zero. So you're really taking the interval a minus zero comma a plus zero, like so. And then in the second case, we also have a radius of convergence of infinity, r equals infinity there. That's because this interval actually looks like a minus infinity comma a plus infinity, like so. And so every power series has a radius of convergence, and every power series has an interval of convergence. And the importance of this interval of convergence is it's the domain of our power series. And we typically use the ratio test to help us determine the radius of convergence. And then what about the endpoints? Well, we have to kind of play a little bit of street fighting series to determine exactly what is going to happen at the endpoints, like we saw in the previous slide.