3Speak - YouTube - Telegram - Notes - Playlist - Sequences and Series - MES Links In this video, I go over the geometric series, which is the sum a + a r + a r^2 + ..., and show that if the absolut...
Transcript
Example 1. So an important example of an infinite series is the Geometric Series. So if we have something like A plus A, yeah, plus A.R, plus AR2, and then going further, plus dot, dot, dot, plus A.R. So this one initially we have, there's AR0. So here's going to be an AR0. This one's AR1, etc. So 0, 1, but this is the first term, so 0, the second term is 1, third term is 2, 4th term is 3. So in other words, we have here, ARN minus 1, plus dot, dot, dot, where the n is just the nth term. So this first term is 0, second term, is 1. So let's see if to subtract by 1, equals to summation, like this, or sigma, infinity of top, n equals to 1 of A, R, and minus 1. And also for this geometric series, I was right, A is not equal to 0. Otherwise, you just get LL0. It'll be all zero everywhere. Yes, and note that each term is obtained from the preceding one by multiplying it by the common ratio R. So this one is A, R0, then you multiply by R, then you multiply by R again, so R squared, R again, R cubed, et cetera, all the way further and further. So note that we have already considered the special case where A equals 1 over 2, and R equals to 1 over 2. Yeah, earlier in this video, and that was one from above where we had 1 over 2 plus 1 over 4 plus 1 over 8, etc, or 1 over 2. So now when we started off with A as 1 over 2, and then we always just multiplied it by 1 over 2, so we had 2n like that. So that was an earlier one. So that was a geometric series that went over, where it equals to 1 here. So this one goes to 1, and this is still a geometric series, where A is equal to 1 half, and then the common ratio is 1 over 2. Let's go further. So now, if we have the case, if R is equal to 1, then these are all just 1. So we just have A plus A plus A plus A. So in other words, then SN equals to A plus A plus A, all the way plus A. In other words, we're going to have n times A, because we're going to have n terms. These are going to be n terms. So end terms. Like that. so we have n terms so n times a equals n a and this just approaches plus or minus infinity depending on the value of of a if it's negative or positive etc so since limit as n approaches infinity or write this better right here so limit as n approaches infinity of s n doesn't exist because it keeps going, plus or minus infinity, there's no actual number, or a real number. The geometric series diverges in this case. But if R is not equal to 1, what we end up having is, let's say it's not equal to 1, and then we're going to have SN equals to all A, plus A, just write this down, A.R, plus A, R squared, plus A, A, R cubed, and keep going at plus dot, dot, plus, and then A, R, and minus 1, like that. And now that we wrote it like this, what we can do is actually a pretty ingenious way, and now what I'll do is, write note, or just type it out, note that by multiplying both sides by R, we get, it's actually pretty amazing, so you get RSN equals to, so this is going to be, a times r so a r but i'm going to do i'm going to space it out over here so we'll have an a r just so it lines up with this to that so it lines up like that so a r and then we have uh yeah so a r cubed plus a r a r to the four So, oops, I got this off. This is, yeah, this is A-R, move it over, it's going to be A-R on this side. A-R times R, it's going to be A-R squared, and then this is going to line up. So notice the pattern, which is absolutely amazing. So we get plus, dot, dot, dot, plus. So all I'm doing is shifting, and this is going to be now, or we're going to eventually get A-R-N-N-minus -1, but then A-R-N-1 times by R, it's going to be, well, plus an A-R-N. So all it was shifted. shift it across there, just so it lines up. So then what we could do is subtract all sides, and you see these all just cancel. These are all just cancel, and all we're going to be left with is this A and this A-R-N, when you subtract it, which is absolutely fascinating, fascinating. Now it's real mathematics. Yeah, so subtracting these equations, we get SNN minus R-S-N, which equals 2, well, we're going to be left with A. everything else cancels, a minus, and then this is A-R-N. And now what we want to do is get the partial sum, SN, so subtract this, or factor it out, S-N is 1 -R, equals to right here and factor out the A, so we have A, 1-R -N, like that. Then move this over, so what we end up having is, we get S-N, the partial sum of the geometric series, equals to A, times the a times 1 minus r n like that over 1 minus r which is absolutely fascinating stuff Yeah, so that's a partial sum, but we're interested in the infinite sum, or just the sum of the full series. So, and also note that if R is between negative 1 and 1, so this is just when R is not equal to 1, and this is, if we have R between negative 1 and 1, recall from example 10 of my video on infinite sequences, that the power function right here r the power of n approaches zero as n approaches infinity if r is a fraction or just in between one and negative one or absolute value is less than one so here is the link to the steamit article there and a screenshot from a calculus book but anyways so remember just to recap there the results of example 10 in that video already summarized for future use as follows feature use of using it right now recall that the sequence r to power of n is convergent if r is between this between it's greater than negative 1 and less than or equal to 1, and divergent for all other values of R. So limit as n approaches infinity of r to power of n is equal to 0 if it's between these. So you could tell it, yeah, it's a fraction, so it's going to be higher and higher values of n. You're just going to be dividing it by more and more. For example, if you have 0.5 squared, this just equals to 1 over 2. square equals 1 over 4 squared, which is, yeah, which equals 1 over 4, which is less than 1 over 2. So it gets smaller and smaller, so it makes sense. And it's approaching 0. And then it equals to 1, if r equals to 1, because r the power of, I mean, 1 to power of n, or 1 power of anything is always 1. So that's what we have, if it approaches 0, so thus we could take the limit here of this partial sum, so limit as n approaches infinity. of s n equals to limit as n approaches infinity of now we just plug this in partial sum a 1 minus r n over uh 1 minus r where this uh infinity right here is this goes to zero because that's uh as above so then this is one times 1 is just 1. So in other words we just get a 1 over R is the full sum of the series of that as it goes to infinity. Just fascinating stuff. Yes, so thus, when the absolute value of R is less than 1, the geometric series is convergent, and its sum is A divided by 1 over R, or A is the initial value, and then divided by 1 minus the common ratio. So if R is less than and equal to negative 1, or r is greater than 1, then the sequence are power of n is divergent and also from the results of the previous video. Previous videos example 10, so here it's divergent for all other values outside of this 1 and negative 1 there, then it's going to be well divergent. So in other words, this goes to infinity. Yes, so hence you're dealing with an infinity value, and that's going to be divergent. So, thus by the above equation, the limit, the limit n approaches infinity of s-N does not exist for other values of R there. Therefore, the geometric series diverges in those cases. If it doesn't exist, you just state it's diverges or divergent. So we summarize the results of example 10 as follows. Yeah, because I'll be using it throughout this video. So summary of example 1, the geometric series, and then we'll write it as this, sigma or summation as n equals to 1 of A r, n times r, oops, a, or like that, n to the, I mean, r to the power of n minus 1. It's been a mouthful there. So a plus a r, plus a r squared, plus dot, dot, dot. Yeah, so this geometric series is convergent if the absolute value of r is less than 1, and its sum is, and then its sum is, let's write this down in infinity, n equals 1 of A, R and minus 1, equals 2, yeah, it just equals to a over 1 minus r, and this is again for r is less than 1. The absolute value of r is less than 1. And yeah, if the absolute value of R is greater than or equal to 1, the geometric series is conversion. And remember if it equals to 1, this is, yeah, if this RN equals to 1, then all you're going to be doing is adding over here A plus a plus a plus a plus a because R is equals 1. And you get n times a and it goes plus from minus infinity. So yes, it's divergent. So note that in words, the sum of a convergent geometric series is, well, just A over 1 minus R. In other words, first term divided by 1 minus common ratio. And a geometric demonstration of the geometric series is shown below, hence the name geometric. Yes, so notice this geometric demonstration right here. So if we have A, so the first term, So this one right here is AR0, this is equals to first term, so A1. This is AR1, this equals to A2, the second term, and so this is the full length of it from here to here. Then you have AR2, this is a third term, then you have here is a... fourth term, and then the full height right here, the full length is S, that's the full summation, and it's called geometric here, because if we take this A, and just flip it over to this side, we have basically a square like this, so there's a square, and then here we have also have a square like this, A-R, and likewise, we have a square, and A-R, so it keeps getting smaller and smaller, and then this height right here, so this is full length A, so here's A, and this is the square than a minus a r because it's full length as a is over there so now I write that because yeah because if the triangles are constructed as shown and s the sum of the series over there like that then by similar triangles the link to that or we get also there we could use similar triangles or just used just basic trigonometry here so if we take this angle right here I'm going to write this angle as theta or better yet, I'll call that theta, I'll call this, this is just a 90 degrees, that's just a square, so this is going to be 90 across there, this one is going to be phi, I'll call that phi, like that, so in other words, this one is just exactly mirrored on this side, so this is going to be phi as well, so yeah, just recall basic angles, but basically geometry, because these angles, these lines are exactly the same thing, so you can just, this is just mirror image of it across this line, mirror image, so we get phi, so in other words, this is theta, this is 90 degrees, this is 5. So in other words, this is 5, this 90 degrees, this angle right here is theta. So now what I'm going to do is, well, take the tangent. I'm going to take 10 of theta, which equals to recall just opposite, over adjacent, the opposite over adjacent. So we take it for both sides. So initially we have here, theta. So the opposite is A minus AR, and then adjacent is A. So we have equals to A minus AR over A. So again, similar triangles. Do it on this side. These are similar triangles. So we have over here, this equals to, yeah, well, we could take any of these ones. The thing is we want to know what this S is. I'll show you why we'll do that. So then what we'll do is opposite is A, and then the full length is S, like that. So we'll have. equals to because we have theta's exact same here, so opposite of this angle, it's going to be a over S. So now what I'll do is I'll just move this A on this side, and then move the S on that side. What we end up having is it also factor that A out. We get, so let's move that over here, so we get S times a factor that out, A1 minus R, this is A times a over there, is equals to a squared. So it'll do is we'll cross this out, cross this out, and then divide this out. We get, well, S is equal to A over 1 minus R, or the first term divided by 1 minus the common ratio as expected, which is absolutely amazing. So, all right, as expected. and hence why it's called the Geometric Series, and this deserves a hashtag amazing.