 So in the previous video we talked about computing geometric sums, what if instead we wanna work on a geometric series? That is to say, what if we want to add together all the terms in a geometric sequence? We'll do all of them, right? Once it's a sequence, there's infinitely many terms right here, so we get this infinite sum. We're gonna take A plus AR plus AR squared plus AR cubed all the way up to, well, we never actually stop. N goes to infinity. Now, in the previous video, we actually found a formula for our S sub N, right? We saw that S sub N will equal A times one minus R to the N over one minus R. This is the general formula. And so what we wanna then consider is that we take the limit as N goes to infinity. That is, we're looking for the limit of this partial sum sequence. Well, let's take the limit as N goes to infinity here. Well, the calculation's gonna go forward in the following way. A is a constant, we can take it out. So we get A times the limit of one minus R to the N over one minus R, because again, this is the limit as N goes to infinity. A doesn't change based upon that. And then the next part is the one minus R, likewise, is constant with respect to N. So you get A over one minus R times the limit as N goes to infinity here of one minus R to the N. But in the next part, again, breaking things up a little bit more, you get A over one minus R times, well, the limit of one is just gonna be one here. So you get one minus the limit as N goes to infinity of R to the N. And so this is the critical junction right here. This limit will be, this sequence will be convergent only if this limit exists right here. The limit as N goes to infinity of R to the N. And there's kind of some options going on right here. Well, what can we say about R to the N? Well, if R's too big, this is gonna go off towards infinity, right? So if like R was two or four or three or pi, those powers are just gonna get bigger, bigger, bigger, bigger, bigger, bigger, bigger. And that's gonna go off towards infinity. So if, so this limit here, so this limit as N goes to infinity of R to the N, this could itself become infinity, right? This is gonna happen if R is like big, right? And this gets actually, it's gonna be, if it's greater or equal to one here, if R is bigger than one, that thing is gonna go off towards infinity, right? What else can we say? If it's a small number though, like if it's one half or one third, what's gonna happen as you start taking powers, this thing is gonna get some smaller over time. So like one half, the next power would be one fourth, then one eighth, then one sixteenth, then one 30 second. These are gonna get smaller, smaller, smaller, smaller, going towards zero in fact. And so in that situation, if you have a small number, makes like slightly amendments to what we had before, if the absolute value of our R is greater than or equal to one, this thing is gonna explode here. I guess they should just say, if it's greater than one, you're gonna explode. And then this would equal zero if our absolute value is less than one. So it takes small numbers. Negative's not a big deal either here. If you took negative powers of one half, that is, I'm sorry to say, if you take powers of negative one half, that thing's also gonna go off towards zeroes right here. And so this number, as you take the limit, will go to zero, like you see right here, only if the absolute value of R is less than one, in which case then the infinite sum will equal this A over one minus R. So you end up with the sum K equals one to infinity of your geometric sequence A times R to the K minus one. This will equal A over one minus R. Now, you'll notice I said that the absolute value of R has to be strictly less than one. If the absolute value of R is exactly equal to one, things can get a little bit different, right? If, we'll just try and do those separately, right? What if R was equal to one? Well, this would then look like our infinite sum would then look like A plus A plus A plus A plus A. Right, continue on, right? And so unless A was zero or something, this is gonna equal, this will equal plus or minus infinity based upon whether A is positive or negative. So that's not a possibility either. And then another issue is if you took negative one, R is negative one, your sum would look something like the following and just for simplicity sake, I'll say A equals one. You end up with one minus one, plus one minus one, plus one minus one, plus one, let's see, plus one minus one, plus one minus one, et cetera. And so this, it turns out this series right here has confused people for a long time because some people all over the following mind says like, oh, I see what's going on here. If I take one minus one, that's a zero. If I take one minus one, that's a zero. One minus one, that's a zero. One minus one, that's a zero. If you have an infinite number of zeros, that adds up to be zero. So this thing is, this thing is convergent to zero, right? Wrong. Let me show you one slight modification of this. If we were to do it this way, we kind of redo the parentheses here, you're gonna have one, so this would be one plus, negative one plus one, which is zero, plus negative one plus one, which is zero, plus negative one plus one, which is zero, you get an infinite number of zeros, right? In which case, this you would get one plus zero, which is actually equal to one. And so one way of associating the parentheses gives you a sum of one, another one gives you zero, which one is it? It can't be one and zero, right? That's actually because this series here is divergent. This sum does not add up to be anything. It doesn't add up to be one, it doesn't have to be zero because it adds up to be nothing. It's divergent, there's no limit to this sequence here. And well, this will make a little bit more sense as we talk about the tests for divergence a little bit later, but like I said, this series has confused people for a long time because some people said it's zero, some people said it's one, in which case then you erroneously might say something like zero equals one. There was in fact middle age clerics that try to make, they did these philosophical proofs of the existence of God, and some of them actually based it upon this argument here that God must exist because we created something from nothing and whether one believes in Deity or not will be a topic I'm not gonna touch into you right now, but this proof of such existence of Deity is flawed because it's based upon a flawed calculus notion. Zero doesn't equal one, this series is divergent and their error seems to come from the convergence of that statement, the belief convergence. And so if we wanna find the following geometric series, I should say, I mean a series is a sum, but it's an infinite sum. So the word series is a little bit better here. Let's find the geometric series right here. And so I claim this is a geometric sequence. Let's look at it real quick. The first term would be two, that's easy to see because it's just the first term right there. What about consecutive terms? If I take four thirds and I divide it by two, that equals two thirds, this is my candidate for R. And if I look at eight ninths divided by four thirds, you're gonna see that's likewise equal to two thirds. And as those are the only three terms they tell us and they claim it's geometric, we will have to take it on Gospel there and see that the constant ratio has gotta be two thirds. And so if we take the sum of this thing by the formula we saw before, right? The infinite sum, take the sum of a geometric sequence A times R to the N minus one as N goes from one to infinity here. This will equal A over one minus R. You just take the sum of all the geometric terms there. And so this is gonna look like the first term, which is two over one minus the common ratio, which is two thirds. And this then equals, well, one minus two thirds is equal to one third. And two divided by, two divided by a third is equal to six, like so. And this gives us, in fact, the sum of this geometric series. It adds up to be six, which is quite incredible. The sum of infinite numbers adds up to be six. And this kind of puts a lot of students back at first, like how in the world could an infinite sum be something finite? But we've actually been doing that for a long time. Integrals are infinite sums as well. It's an infinite Riemann sum, which again is slightly different than a series here, but a number of infinite things can in fact be finite. Another sort of example you can kind of think of is imagine we have a rope, which is exactly equal to one foot long. Well, if we were to cut it into two pieces, well, then we have two ropes, which are of one half length. Well, what if we take the second rope and we cut it in half, then we'll get a rope of one half, then a rope of one quarter, and then the other one is one quarter. Well, if we cut that one in half, we'll get two ropes of one eighth. The total length is still one. If we cut that last one in one half, we get one sixteenth, and then another sixteenth, and then we keep on doing that. We keep on cutting the last rope in half, half, half, half, half, half, and we do that at infinitum. Then it turns out that we will have an infinite number of pieces of rope, but each rope, when you add them all together, the length is still equal to one. And this is sort of a concrete way of trying to envision the idea that one half plus one fourth plus one eighth plus one sixteenth. If we take up all the powers of two, this will add up to be one. And this is, of course, is also, again, I kind of alluded to this earlier. This infinite convergence is sort of a thing that baffled the earlier scholars and such, right? If you look at the motion paradox of Zeno, known as Achilles and the tortoise, he tries to argue that existence, as we know it, is flawed because motion's impossible because motion would acquire an infinite sum. And Zeno then concluded that an infinite sum doesn't exist. Well, that, I would say, is a flawed argument because Zeno wasn't using proper notions of calculus here. An infinite sum actually can equal a finite number. That was the flaw in Zeno's argument there. So I'm not willing to say that motion in physics, as we know it doesn't exist. Let's look at one more example here of a geometric series here. So is the series n equals one to infinity of two to the two n times three to the one minus n convergent by virgent? Well, again, I kind of let the cat out of the back here. We're gonna represent this as a geometric series, in which case it's very easy to check whether a geometric series is convergent or divergent. And it requires rewriting things a little bit. This expression right here, this two to the two n times three to the one minus n, I'm gonna rewrite this using some exponent rules, right? So two to the two to the n, I can rewrite this as two squared times n using my usual exponent laws. And three to the one minus n, I can write as three to the first times three to the negative n, like so. N will range from one to infinity here. And again, rewriting these things a little bit more, we're gonna get the sum of four to the n on top times it by three over three to the n on the bottom, where n goes from one to infinity here. And so what we end up with is three times the sum where n ranges from one to infinity. And we're gonna have a four thirds to the n right here. And so this is where we can actually see evidence of a geometric sequence. We have our, we have some constant value right here times by four thirds. But in particular, we see this exponential expression right here, this exponential expression, four thirds to the n. So this is really what tells us we have a geometric sequence. And so what one has to do is just compare the ratio here. The ratio is gonna be the base of this exponential growth. The ratio will be four thirds. And this four thirds is too big, right? A geometric series will be convergent. A geometric series will be convergent if and only if your common ratio is, adaptive value is less than one. We say the ratio is small. Otherwise the series will be divergent. The geometric series will be divergent if and only if our ratios, absolute value is greater than equal to one. That is to say it's a big ratio. And this is the case that we're in right now. This, our ratio here is big because it's four thirds. And therefore we would conclude that this is a divergent geometric series. This thing doesn't necessarily add up to anything. It doesn't add up to a number. If we want us to describe something to it, we could say that it's equal to infinity. But if the series adds up to be infinity that still is the example of a divergent series.