Infinite Geometric Series & Intro to Limits in Calculus - Part 1 - [18] - By Math and Science
Transcript
00:00 | Well , welcome back . The title of this lesson | |
00:02 | is called the Infinite geometric series . This is part | |
00:05 | one of several . I'm really excited to teach this | |
00:08 | lesson because of a couple of reasons . One it | |
00:10 | has a kind of a counterintuitive lesson , it doesn't | |
00:13 | make sense at first , but I will absolutely make | |
00:15 | have it make 100% sense by the end here . | |
00:17 | But more importantly , it's because it dovetails very , | |
00:21 | very nicely into calculus . So here we're kind of | |
00:23 | at an algebra pre calculus level , but we are | |
00:25 | going into calculus soon and so I get to introduce | |
00:29 | some calculus topics here . I do not believe in | |
00:32 | babying you , I do not want to introduce too | |
00:34 | much calculus too early . However , when there's a | |
00:36 | good opportunity then then I'm not gonna pass that opportunity | |
00:39 | up and that's what we have here . So you're | |
00:41 | gonna learn a little bit of some sort of very | |
00:42 | elementary calculus concepts . The biggest one here that you | |
00:46 | need to kind of wrap your brain around is when | |
00:48 | we talk about the infinite geometric series , we're talking | |
00:51 | about a series of terms that are added together . | |
00:54 | The geometric series , right ? But an infinite number | |
00:57 | of them , not 10 of them , not 50 | |
00:59 | of them , not 1000 of them , not a | |
01:00 | million of them , not 20 quadrillion of them , | |
01:03 | literally an infinity of them . So the first time | |
01:06 | you think about that , it should kind of blow | |
01:07 | your mind . How can you add an infinity of | |
01:10 | things together ? I mean that means there's an infinite | |
01:11 | number of things . That means I can continue adding | |
01:14 | them till the end of the universe and I'm still | |
01:16 | not done adding them . So how is it possible | |
01:19 | that I can add up these guys at all ? | |
01:22 | And furthermore in this lesson , we're gonna find out | |
01:24 | that when you add up terms of a geometric series | |
01:27 | under a certain condition , I'll tell you in a | |
01:29 | minute , then not only are you adding an infinite | |
01:32 | number of terms , but if you are allowed to | |
01:33 | do that , you're actually going to get a number | |
01:35 | a single number . How can that be ? If | |
01:37 | you add an infinite number of things down ? Shouldn't | |
01:40 | the answer be infinity ? If I'm adding forever , | |
01:42 | shouldn't be answer B infinity . In other words , | |
01:45 | I always never get to the answer Actually , that's | |
01:47 | not true and that's super important for you to conceptualized | |
01:51 | because calculus is completely based on this idea , the | |
01:54 | whole branch of calculus that you learn the second half | |
01:57 | of calculus one is all about adding up an infinite | |
02:00 | number of things . It's crucial to our modern ideas | |
02:03 | of engineering and math and science that we use all | |
02:06 | the time to calculate things . So , we're kind | |
02:08 | of inching our way to those ideas in this lesson | |
02:11 | . All right . So we need to back the | |
02:12 | truck up a little bit and talk about the infinite | |
02:14 | geometric series . What I'm gonna do is write down | |
02:16 | the some the actual finite sum of an infinite geometric | |
02:20 | series . Going to write it down . And then | |
02:22 | I'm gonna back up the truck and take you from | |
02:24 | the beginning so that you understand exactly where this equation | |
02:26 | comes from and then we're going to apply it to | |
02:28 | lots of problems . The good news is it's very | |
02:31 | easy to understand . And the problems are very , | |
02:33 | very simple . But I do need you to watch | |
02:35 | this lesson entirely so that you will be with me | |
02:37 | all the way to the end . All right . | |
02:39 | So , what I wanna do first , like I | |
02:40 | said , is I want to give you the punch | |
02:42 | line first . Okay . It turns out that I | |
02:44 | can add up and find the sum of an infinite | |
02:51 | an infinite set of terms in terms of a geometric | |
02:58 | series . So when you see the word infinite in | |
03:01 | front of geometric series , it means the terms never | |
03:04 | ever end Now . In order for a geometric series | |
03:07 | to be to be when you add them up to | |
03:09 | actually get a number out of it , a single | |
03:12 | number . Okay , is a very important constraint . | |
03:15 | Remember in this geometric series we have what we call | |
03:17 | the common ratio . So we have to say that | |
03:20 | if the common ratio , the absolute value of it | |
03:22 | is less than one . Okay then the following is | |
03:28 | true . The some Of notice I didn't say s | |
03:32 | . s . n . I mean the sum of | |
03:33 | all infinite , all infinite number of those terms is | |
03:37 | equal to the first term divided by 1 -2 common | |
03:40 | ratio . Right , so let me back up a | |
03:43 | little bit and talk about this . When I say | |
03:45 | if the absolute value of our is less than one | |
03:48 | , I need to talk about what this means . | |
03:50 | What this means is the following . It means That | |
03:55 | the common ratio has to be greater than -1 . | |
03:59 | Okay . And The common ratio has to be less | |
04:03 | than one . Okay . It's very confusing when you | |
04:07 | see it written like this , but almost every book | |
04:08 | is gonna they're not gonna write it like this . | |
04:10 | Everybody can understand this art has to be greater than | |
04:13 | negative one and are has to be less than one | |
04:15 | . That makes sense to me , bigger than negative | |
04:17 | one . Right ? And less than one . In | |
04:19 | fact , if I put this on a number line | |
04:20 | , what I'm basically saying here is if this is | |
04:22 | the common ratio and this is zero and this is | |
04:24 | one and this is negative one . What I'm saying | |
04:26 | is that common ratio has to be here . It | |
04:30 | has to be a fraction a negative fraction less , | |
04:33 | a little bit bigger than negative one and a fraction | |
04:36 | a little bit less than one . In other words | |
04:38 | , the common ratio for this to work has to | |
04:40 | be like one half or three quarters or 30.99999999 Just | |
04:45 | a little bit less than one . Or it can | |
04:46 | be negative one half or negative three quarters or negative | |
04:48 | 30.9999999 Just like this , If you make the common | |
04:52 | ratio , anywhere in here where it's a fraction smaller | |
04:56 | than one , Either positive or negative , then you | |
04:58 | can find a some of the series and when I | |
05:01 | say that some of the series , I mean a | |
05:02 | sum of all infinity terms that never ever end . | |
05:05 | That's what's so crazy about it . But it's absolutely | |
05:08 | true . I'm gonna show you an example why that's | |
05:10 | the case and that some is going to end up | |
05:12 | becoming the first term divided by one minus this common | |
05:15 | ratio . Now , one thing I need to point | |
05:18 | out before we get too much farther is that when | |
05:21 | you have a constraint , like the common ratio has | |
05:24 | to be bigger than one in less than one , | |
05:27 | all it's telling you is the common ratio has to | |
05:29 | be a fraction less than one but either positive or | |
05:32 | negative . Remember , common ratio is telling me how | |
05:35 | to find the next term in the next term in | |
05:37 | the next term . If the common ratio is bigger | |
05:39 | than one Outside of this region , if the conversation | |
05:42 | is five , five for instance , that means I'm | |
05:44 | multiplying by five and the terms are getting bigger and | |
05:46 | bigger and bigger . If I add up an infinity | |
05:48 | of terms that are getting always bigger , then of | |
05:50 | course I'm gonna get infinity . I'm not going to | |
05:52 | get an actual some it's going to be no some | |
05:54 | it's gonna become infinity . So the only way you | |
05:56 | can get a some actual number is if the common | |
06:00 | ratio is a fraction , that means the terms are | |
06:02 | getting smaller and smaller and smaller . That's the only | |
06:04 | way it works . If the common ratio is one | |
06:06 | half , for instance , that means I'm multiplying by | |
06:09 | one half to get each next term , so multiplied | |
06:11 | by a half , multiplied by a half and so | |
06:13 | on . So all the terms are going down and | |
06:15 | that's when you can say the geometric series sum is | |
06:18 | equal to a number . That's why the common ratio | |
06:20 | has to be in that window . The negative if | |
06:23 | it's a negative fraction , it's the terms are still | |
06:25 | gonna get smaller , they're just gonna alternate signs but | |
06:28 | they're still gonna get smaller . If the ratio is | |
06:30 | over here , the terms will be positive and they'll | |
06:32 | get smaller if the common ratio is anywhere outside of | |
06:36 | this region , like negative to our positive to the | |
06:38 | terms will get either bigger or they'll get bigger . | |
06:41 | But alternating bigger , the negative are means it's gonna | |
06:44 | multiply by negative so it's gonna get bigger . It'll | |
06:46 | get alternating bigger , it'll go out of control . | |
06:49 | So you won't have a some When you see it | |
06:51 | written like this , what it's saying is if you | |
06:53 | were to put like let's say negative .5 in here | |
06:56 | , take the absolute value . That's less than one | |
06:59 | . If you put a positive .5 in here , | |
07:01 | take the absolute value , it's still less than one | |
07:03 | . So writing it like this is just a shorthand | |
07:05 | way of saying this . Okay , so enough of | |
07:08 | that . I don't want to talk about that anymore | |
07:10 | . That is the punchline . That's what the some | |
07:12 | of the geometric series is . What I want to | |
07:14 | do now is show you why and not just with | |
07:17 | a simple little proof . I want to give you | |
07:18 | a real example . So what I want to do | |
07:20 | is let's consider a very easy to understand geometric series | |
07:25 | . So consider The following geometric series . We've looked | |
07:29 | at this before . one half plus 1/4 plus 1/8 | |
07:35 | plus 1/16 plus dot dot dot . When I say | |
07:39 | dot dot dot , that means there's an infinity of | |
07:41 | terms after this term would be won over 30 to | |
07:44 | 1 32nd , then it would be won over 64 | |
07:47 | . There will be 1/1 28 . There'll be 1/2 | |
07:50 | 56 . It would go on and on and on | |
07:51 | forever and ever and ever . But notice that these | |
07:54 | terms are getting smaller . Why are they getting smaller | |
07:57 | ? Because what is this common ratio ? How do | |
07:59 | I go from this term to this one ? I | |
08:02 | multiply by one half . One half times a half | |
08:04 | is 1/4 . How do I go from here ? | |
08:06 | Multiply by one half . 14 times a half is | |
08:09 | 1/8 . How do I go from here to here | |
08:11 | ? Again ? I multiply by one half this times | |
08:14 | this gives me this . So the common ratio here | |
08:16 | is one half . Right . And that means that | |
08:20 | the common ratio is less than one , the absolute | |
08:23 | value of it . So if you put a one | |
08:24 | half in here , take the absolute value . It's | |
08:26 | less than one . Check . That means that we | |
08:29 | already know ahead of time that this series is going | |
08:32 | to have a some and if I wanted to calculate | |
08:34 | the sum , which I'm not gonna do yet , | |
08:36 | but we will do it . All I would have | |
08:38 | to do is put the first term in here , | |
08:39 | one minus the uh the uh the common ratio that | |
08:44 | we just figured out and we can calculate the something | |
08:47 | like that . I don't want to do it yet | |
08:48 | . I want to show you why it works okay | |
08:50 | , but we know that the common ratio criteria is | |
08:53 | correct . In other words , it's one half the | |
08:55 | common ratio is right in here and that means these | |
08:58 | times are getting smaller and that means this is going | |
09:00 | to have a some . And by the way , | |
09:02 | when we say a series has a sum , an | |
09:04 | infinite series has a some , we say that the | |
09:06 | series converges . When you say the series converges , | |
09:11 | it means it has a some , even though there's | |
09:12 | an infinite number of terms . When you say the | |
09:15 | series diverges , that means there's no some , it | |
09:17 | just goes off to infinity and lots of series also | |
09:20 | diverge as well . So let's take a look at | |
09:23 | this in more detail . That's what I want to | |
09:25 | do next . Yeah , let's look at the partial | |
09:28 | subs . Remember that we talked about that partial sums | |
09:34 | . What is the s sub one ? That means | |
09:36 | the sum of the first term . Well if you | |
09:38 | look at the first term , that's only one half | |
09:39 | . So there's nothing to really add . Okay , | |
09:42 | but if you uh Were to convert this fraction to | |
09:45 | a decimal , it's exactly equal to 0.5 . Okay | |
09:50 | , now let's look at the second partial sum , | |
09:52 | the sum of the first two terms . And that | |
09:54 | means it's gonna be one half plus 1/4 because that | |
09:57 | is the first two terms . How do I add | |
09:59 | these ? Okay , just to make sure everybody's on | |
10:01 | the same page I could say well I've got one | |
10:03 | half , I'm adding to it . 1/4 . I | |
10:06 | need a common denominator . So I multiplied by four | |
10:08 | of four . And so what am I gonna have | |
10:10 | actually ? It's not gonna be for before . Sorry | |
10:12 | about that . Working a little bit too fast . | |
10:14 | It's gonna be I can multiply by 2/2 . So | |
10:16 | this will be 2/4 Plus 1 4th . What do | |
10:20 | you get here ? 3/4 . Okay . And as | |
10:24 | you have 3/4 is the answer ? The exact value | |
10:26 | is 0.75 . Because 3/4 you all know is 0.75 | |
10:31 | . Let's take a look at the third , partial | |
10:33 | sum . What will it be ? One half plus | |
10:36 | 1/4 plus the third here is 1/8 . Okay . | |
10:41 | How do you do all of this ? Will you | |
10:43 | need a common denominator and you'll have to find a | |
10:45 | common denominator with an eight and an eight here and | |
10:48 | I'll just do it one more time to make sure | |
10:49 | we're on the same path here we have one half | |
10:52 | 1/4 18 So to get a common denominator multiply here | |
10:57 | by 4/4 . Multiplied here by 2/2 . So what | |
11:01 | do I get I get here ? 4/8 . Okay | |
11:05 | here I will get to 8th and here I have | |
11:10 | the 1/8 . So when I go back over here | |
11:12 | I'll put an equal sign right below . So four | |
11:15 | plus two is six plus one is 7/8 . That's | |
11:18 | the common denominator here . And whenever you Um convert | |
11:21 | this to a decimal 7/8 , you're going to get | |
11:24 | 0.875 . All right , Bear with me . We're | |
11:29 | actually almost done . Let's take a look at the | |
11:31 | fourth , partial sum . One half plus 1/4 plus | |
11:36 | 1/8 plus . What is that ? Fourth term ? | |
11:38 | 1/16 . All right . We'll get the common denominators | |
11:42 | . I'm not gonna do it all . But you | |
11:43 | have a common denominator of 16 that you could get | |
11:45 | multiply . All these guys add them all up . | |
11:48 | What are you gonna get ? You'll actually get 15 | |
11:50 | 16th ? I encourage you to do that on a | |
11:52 | separate sheet of paper and when you crank through this | |
11:54 | and get a decimal , what do you get ? | |
11:56 | 0.9375 I think you can see what's happening here . | |
12:01 | But just to bring the point home let's skip down | |
12:04 | . So there's 1/5 partial sum in the sixth and | |
12:06 | the seventh . Let's add the 1st 10 terms up | |
12:09 | the 10th partial sum . So what will you have | |
12:12 | ? It'll be one half plus 1/4 plus 1/8 plus | |
12:17 | 1/16 plus dot dot dot . And then well , | |
12:20 | let's put a plus on here . 1/10 24 . | |
12:25 | Because the last term when you multiply by a half | |
12:28 | 10 times is gonna be 1/1 to 4 . If | |
12:30 | you get a common denominator of all these and add | |
12:32 | them up , You will get 1023 Over 1024 . | |
12:38 | You see this is really , really close to one | |
12:41 | . In fact , what do you get here ? | |
12:43 | It's going to be 0.999 02 . And this is | |
12:49 | I'm gonna put approximate because this is rounded What have | |
12:54 | we done here ? We started out and said , | |
12:56 | here's an infinite geometric series is an infinite number of | |
12:58 | terms . I'm telling you ahead of time , this | |
13:01 | has a converges . In other words , you can | |
13:03 | add an infinite number of terms and have actually equal | |
13:06 | a number , right ? If we only add the | |
13:09 | first term up . In other words only consider the | |
13:11 | first term , we get .5 if we had the | |
13:13 | first two terms , this is the answer . If | |
13:15 | we had the first three terms , .875 , we | |
13:18 | had the first four terms . .9375 . The 1st | |
13:21 | 10 terms . 0.99902 What do you think is gonna | |
13:24 | happen if we had the 1st 1000 terms ? Because | |
13:27 | this is an infinite series . We we got to | |
13:29 | the 1st 10 terms and already got a number really | |
13:31 | close to one . What's going to happen if we | |
13:33 | get to the 1st 15 , 20 terms ? What | |
13:35 | about 1000 terms ? What if we take a million | |
13:38 | terms ? What if we consider 20 quadrillion billion gazillion | |
13:41 | ? That's not even a number . But if we | |
13:42 | look at all those terms , what's gonna happen is | |
13:44 | you're gonna get .999999 , you're gonna get really , | |
13:47 | really , really , really close to a number . | |
13:49 | This series converges . You consider more and more and | |
13:52 | more terms trying to get as close to infinity as | |
13:55 | you can . And all that happens is the sum | |
13:57 | gets closer and closer and closer to a finite number | |
14:00 | . That finite number is actually going to be equal | |
14:02 | to one . So to put that in words , | |
14:06 | okay , this is where the calculus part comes in | |
14:08 | . What we say is as the number of terms | |
14:11 | that we consider approaches infinity , this arrow means you | |
14:15 | can never get to infinity . You can just consider | |
14:17 | more and more and more terms like I was telling | |
14:19 | you verbally but when we consider more and more more | |
14:21 | terms then the uh impartial some when you put infinity | |
14:27 | down here uh is going to approach the number one | |
14:30 | . Will it ever quite get to the number one | |
14:32 | ? Well , no , because we can't ever calculate | |
14:35 | an infinity of terms . But conceptually the closer we | |
14:38 | get , it always approaches closer and closer and closer | |
14:41 | to a finite number . The number one . So | |
14:43 | theoretically if you could add an infinity of terms , | |
14:46 | it would exactly equal the number one . That means | |
14:49 | the series converges . If it were to go up | |
14:53 | to infinity when we add up these numbers than it | |
14:55 | would diverge . So we say that this thing converges | |
14:57 | and so in the language of calculus , Right . | |
15:01 | So in the language of calculus , whoops , if | |
15:04 | I can spell calculus correct , This is what we | |
15:07 | say . We say that this thing has what we | |
15:10 | call a limit . When we let the limit as | |
15:12 | the number of terms go to infinity of the partial | |
15:16 | , some of the number of terms . In other | |
15:18 | words we put in is one inches five inches three | |
15:21 | inches 35 is 100 is a million and so on | |
15:25 | . This limit equals one . If we let in | |
15:28 | actually approach infinity , gets closer and closer and closer | |
15:30 | . So we say that when in becomes infinity and | |
15:33 | in fact it can never become infinity . But as | |
15:35 | it approaches infinity , this limit becomes one . Okay | |
15:39 | , So we'll kind of put this guy right here | |
15:41 | and we say this series , we say it converges | |
15:48 | in your algebra book . If you're looking at an | |
15:51 | algebra book , it'll probably won't say that . It'll | |
15:53 | probably say the series has a some or there's a | |
15:56 | finite sum to the series . But in the language | |
15:58 | of calculus , you say the series converges in that | |
16:01 | case . And you say that the limit of the | |
16:03 | some of the partial sums as n goes to infinity | |
16:06 | is is equal to one is equal to a finite | |
16:08 | number . So , as an example of when you | |
16:10 | can add up an infinity of things , but get | |
16:12 | a finite number . But notice the only reason it | |
16:14 | works is because these terms are getting smaller . If | |
16:17 | these terms are getting bigger , like if this series | |
16:19 | were 2468 10 , then as I add the infinity | |
16:23 | of them together , of course it's gonna blow up | |
16:24 | , it's gonna diverge . That's what will happen . | |
16:26 | But that's not what's happening when you have a geometric | |
16:29 | series where the terms are getting smaller and smaller and | |
16:31 | smaller . Then we say the series has a finite | |
16:35 | answer . But why exactly does it approach one ? | |
16:39 | I mean obviously we can we can write it down | |
16:42 | , we can see that it approaches one we calculated | |
16:44 | . But fundamentally let's go a different angle and let's | |
16:46 | try to figure out why it approaches one . So | |
16:48 | , I'm gonna put a big y here . That's | |
16:52 | what we're gonna investigate next . I want to talk | |
16:54 | to you about why . So I want you to | |
16:56 | recall the ends partial sum . We talked about geometric | |
17:01 | series already . We said we can add up a | |
17:03 | finite number of terms . And we said that that | |
17:06 | some the finite sum was called S . A N | |
17:09 | . And it was T the first term times one | |
17:11 | minus R . To the power of in over one | |
17:15 | minus R . This is the uh the equation for | |
17:19 | the impartial some that we have derived before . If | |
17:21 | I want to figure out what the first three terms | |
17:23 | of the series is . That's okay . I put | |
17:26 | the common ratio in , put in is equal to | |
17:28 | three . I know the first term crank through it | |
17:30 | and I'll get the some of the first three terms | |
17:32 | . In fact , I could use that equation to | |
17:34 | calculate the second parcel . Some the third the fourth | |
17:37 | to 10th . If you put these numbers in here | |
17:40 | , you will exactly get back the numbers that we | |
17:42 | got by hand here . That's exactly what you would | |
17:44 | get . All right . But let's go through it | |
17:49 | for the case when we're trying to go to an | |
17:50 | infinity of terms with this common ratio that we have | |
17:53 | . So in this case uh what we have is | |
17:58 | the first term in our series . If you go | |
18:00 | back to our series , the first term was one | |
18:02 | half . So we'll put T one is one half | |
18:06 | , one minus the common ratio . What was our | |
18:09 | common ratio ? The common ratio was one half . | |
18:11 | We kept cutting the terms in half every time . | |
18:13 | So the common ratio was one half Power of in | |
18:17 | is still right here . On the bottom we have | |
18:19 | 1 - are we just said the common ratio was | |
18:21 | one half . So what's gonna happen here ? S | |
18:23 | Sir Ben ? The some of the term is going | |
18:27 | to be 1/2 . Yes , Let's go in here | |
18:30 | and say 1 -1 half again to the end . | |
18:34 | Power on the bottom one minus a half . What | |
18:36 | do you get one half ? So what's gonna happen | |
18:39 | is you have a coefficient of one half on the | |
18:41 | top of one half on the bottom . So what | |
18:44 | actually happens here is the 10th partial sum . The | |
18:47 | only thing you have left is this 1 -1 half | |
18:51 | to the power event . What is this equation telling | |
18:53 | you ? This is telling me if I have a | |
18:55 | geometric series that starts with the first term of one | |
18:59 | half and that has a common ratio of cutting every | |
19:02 | term in half each time . In other words , | |
19:04 | it's for our specific series that we just looked at | |
19:07 | on the board , then the sum that you get | |
19:09 | when you consider the first in terms is equal to | |
19:12 | this . So this means that when uh when in | |
19:17 | gets very , very big , right , what's gonna | |
19:19 | happen when in gets big ? This is one half | |
19:21 | to the in power . If it's one half to | |
19:23 | the 10th , what's going to happen ? It's one | |
19:25 | half times one half times one half times one happened | |
19:27 | . You do that 10 times . So when in | |
19:29 | gets bigger and bigger and bigger , this thing gets | |
19:32 | smaller . So as in gets bigger , what happens | |
19:38 | is one half to the end gets smaller . Why | |
19:41 | ? Because it's a fraction raised to an exponent . | |
19:44 | So it's one half times one half times one half | |
19:46 | as it gets bigger , bigger , bigger , then | |
19:48 | this thing gets smaller , smaller , smaller . So | |
19:51 | as in approaches infinity . This some approaches one because | |
19:59 | this thing is getting smaller and smaller and smaller . | |
20:01 | And so the only thing left is one . So | |
20:03 | I just wanted to show you like it's one thing | |
20:06 | to just write the terms out and to say , | |
20:08 | oh yeah , it looks like they're getting close to | |
20:09 | one . That's cool . But it's another thing to | |
20:11 | take a different angle and say this is the equation | |
20:14 | that we already used for the N number that some | |
20:17 | of the some of the first in terms we put | |
20:19 | in the information for our specific series with our specific | |
20:23 | common ratio and our specific first term . And what | |
20:26 | you get out of that is something that looks like | |
20:28 | this and it's very clear that as in gets bigger | |
20:31 | , all that's happening is the sum is getting closer | |
20:33 | to one . That is why the sum is getting | |
20:35 | closer to one over here . Because the equation we | |
20:37 | already learned about to calculate the sums approaches one as | |
20:42 | N gets bigger and bigger and bigger . Now , | |
20:43 | what I want to do is I want to check | |
20:45 | my math here and make sure that I didn't forget | |
20:49 | anything . So I guess the one last thing I | |
20:52 | can say , so we can say that one half | |
20:54 | plus 1/4 plus 1/8 plus 1/16 plus dot dot dot | |
21:00 | . We can say that that's actually equal to the | |
21:02 | number one . But implicit in this is that whenever | |
21:06 | you add the dot dot dot , so you're adding | |
21:08 | an infinite number of things beyond it . Only when | |
21:11 | you consider all infinity of terms does the some actually | |
21:14 | converge to one . If you stop at N is | |
21:18 | equal to three million terms , then it's gonna be | |
21:20 | real close to one , but it's not gonna be | |
21:21 | one . If you consider all infinity of terms , | |
21:24 | which kind of blows your mind then you can actually | |
21:26 | get to one . Okay ? And now you can | |
21:30 | see why it's so critical in the beginning here , | |
21:33 | what did I tell you ? I said the some | |
21:36 | of the infinite geometric series is going to have this | |
21:39 | , we're going to come back to this , this | |
21:40 | is gonna be the sum . But in order for | |
21:42 | the sum to exist at all , the common ratio | |
21:45 | has to be less than one . I mean this | |
21:46 | means less than positive ones are bigger than negative one | |
21:49 | . It has to be a fraction somewhere in this | |
21:50 | region . That's what it's saying . Now , you | |
21:53 | can see actually why that's the case . The common | |
21:55 | ratio went in here . In our case it was | |
21:57 | one half . So when you go through the math | |
22:00 | , the common ratio pops out here and so what's | |
22:02 | going on is if the common ratio is a fraction | |
22:06 | , then when end gets bigger , this term goes | |
22:09 | down down , down , down down as the terms | |
22:11 | get bigger . But if that common ratio we're not | |
22:14 | a fraction , let's say the common ratio were to | |
22:16 | let's say let's say let's put it like a two | |
22:18 | in here instead of one half to the and it | |
22:20 | was like to to the end when the common ratio | |
22:22 | is too big , that means that this term never | |
22:25 | gets smaller , it just gets bigger and then the | |
22:26 | whole thing blows up to infinity and the series does | |
22:29 | not converge . That is why the common ratio has | |
22:33 | to be a fraction to force the terms to go | |
22:35 | down close to zero . Then in that case you | |
22:38 | can add an infinite number of terms together and get | |
22:40 | a number . If the common ratio is outside of | |
22:42 | that of that range , then the terms don't go | |
22:45 | down and then the series does not convert . And | |
22:48 | just to kind of illustrate that a little bit more | |
22:51 | directly . Let's consider another series . Let's consider another | |
22:55 | series . Consider Consider one plus 10 plus 100 plus | |
23:05 | 1000 plus dot dot dot This is a geometric series | |
23:09 | . How do you know well , because to go | |
23:11 | from this term to this term , I have to | |
23:12 | multiply by 10 to go from this term to this | |
23:15 | term . Multiply by 10 . This term to this | |
23:16 | term multiply by 10 . It's a geometric series . | |
23:19 | So you might say , well I'm going to add | |
23:21 | them up and figure out what happens here but notice | |
23:23 | that the common ratio here is equal to 10 and | |
23:26 | that's way outside of of what I said , it | |
23:30 | had to be . The ratio has to be between | |
23:32 | negative one and positive one . This thing is way | |
23:34 | outside of that . And because of that , as | |
23:37 | in approaches infinity is I look at what happens if | |
23:39 | I start adding all these terms , then the some | |
23:42 | approaches infinity as well . It does not converge , | |
23:45 | so does not converge . In other words , there's | |
23:52 | no some because as you include more and more terms | |
23:54 | that some blows up instead of getting um smaller . | |
23:58 | So geometric series can add up to a finite number | |
24:02 | , but only when the common ratio is between negative | |
24:04 | one and positive one . That's the bottom line when | |
24:07 | the common ratios between negative one and positive one , | |
24:10 | the terms of the series are going down and because | |
24:12 | of that there's a finite sum and that finite sum | |
24:15 | is equal to this . Now the last part of | |
24:17 | what I want to do is show you why the | |
24:19 | some actually equals this . So to figure that out | |
24:23 | to prove it . Let's take a look at that | |
24:26 | . Let's take a look at a proof of this | |
24:29 | . Remember the and partial sum we talked about that | |
24:31 | before . It was T one one minus R . | |
24:34 | To the power event over one minus . Are we | |
24:37 | covered this a couple lessons go ? You've learned it | |
24:39 | . It's the some of the first in terms of | |
24:41 | the geometric series . Okay . But I want you | |
24:44 | to consider the case . Consider if n goes to | |
24:49 | infinity right ? We have to consider the first infinity | |
24:52 | terms . Right ? And if that happens , uh | |
24:56 | if Absolute value are less than one , that means | |
25:00 | the common ratios between plus or -1 , right ? | |
25:04 | Then what's gonna happen here in this sum is R | |
25:07 | to the power of N is in that case gonna | |
25:09 | go zero . Why ? Because if the common ratio | |
25:12 | is in the range that I'm saying it has to | |
25:13 | be it has to be a fraction like put a | |
25:15 | one half in here , then one half to the | |
25:17 | end is I let in go to infinity will be | |
25:19 | one half times one half times one half times one | |
25:21 | half . So if you do an infinite number of | |
25:23 | times this term right here goes away . So what | |
25:26 | will happen is it will be something like this S | |
25:30 | n t 11 minus R to the power of end | |
25:34 | over one minus are if all this is true . | |
25:39 | In fact , I can kind of say , well | |
25:41 | let's consider an infinite number of terms and then we'll | |
25:44 | put infinite number of terms here . But what's gonna | |
25:45 | happen if R is a R is a really a | |
25:49 | fraction like we're saying , and you raise it to | |
25:50 | that power of infinity this term just goes to zero | |
25:54 | . In the case , when the common ratio is | |
25:56 | small like that , if the common ratio is bigger | |
25:58 | than one , then I raise it to an infinity | |
26:00 | power , then it's gonna get huge . It's gonna | |
26:02 | go to infinity . But in the case when it's | |
26:04 | a fraction like this , it doesn't it doesn't happen | |
26:06 | . So what do I have left ? The sum | |
26:08 | of everything is going to be the first term times | |
26:11 | one , which means t ones on the top and | |
26:13 | one minus R is on the bottom . This is | |
26:15 | exactly what I told you . The sum is equal | |
26:18 | to . So the reason why the sum of an | |
26:21 | infinite geometric series of terms is equal to this is | |
26:24 | because when you look at the somme that we learned | |
26:26 | before for a finite number of terms and consider the | |
26:29 | case when the common ratio is a fraction , then | |
26:32 | if I let infinity of terms in there , this | |
26:34 | whole entire thing just drops away . it goes to | |
26:36 | zero and all I'm left with is this this is | |
26:39 | kind of the proof of that . You just examine | |
26:41 | the case when the common ratio is a fraction and | |
26:43 | let the number of terms go to infinity . Boom | |
26:45 | . That term drops away all you're left with . | |
26:47 | Is this So what I would like to do now | |
26:51 | is let me see if I have a room over | |
26:54 | there . Yeah , let's do it over here . | |
26:57 | Let's go back to our example that we started the | |
27:00 | lesson with and we started the lesson and we said | |
27:04 | consider the following series . We said what about one | |
27:07 | half plus 1/4 plus 1/8 plus 1/16 plus dot dot | |
27:14 | dot It's an infinite series . And we said we | |
27:18 | did the terms , we know what it comes out | |
27:19 | to be . It comes out to one right ? | |
27:21 | We already said this , but let's apply this . | |
27:23 | We're saying the sum is going to be equal to | |
27:25 | in the case when the common ratio is a fraction | |
27:28 | between plus or minus one , we know it is | |
27:31 | because we've already done that , the sum is gonna | |
27:33 | equal to the first term , which is one half | |
27:35 | divided by one minus the common ratio , which the | |
27:38 | common ratio . Again we've already said is one half | |
27:41 | . So what do you get ? 1/2 over one | |
27:43 | half ? So what does this mean ? It means | |
27:45 | the sum of all this stuff is one half divide | |
27:47 | by one half , which is one . This is | |
27:49 | the same . Some that we calculated in the very | |
27:53 | beginning we took these terms , we literally did it | |
27:56 | by a brute force method of taking more and more | |
27:58 | and more terms , adding them up . And we | |
28:00 | figured out that it approached one and we did a | |
28:02 | bunch of talking and kind of looking at how it | |
28:04 | all works and how to look at that equation and | |
28:06 | how everything works out and it all works out to | |
28:08 | one . But then we say here's a theorem , | |
28:10 | you can use it for any geometric series , if | |
28:12 | you know the first term and you know that the | |
28:14 | common ratio is between plus or minus one , then | |
28:17 | this is the sum . All you need is the | |
28:19 | first term and the common ratio . When you have | |
28:21 | an infinite number of terms , you add them up | |
28:23 | , that's what you get . We apply it to | |
28:24 | the same exact some that we started with to begin | |
28:27 | with and we get exactly the same answer . So | |
28:30 | I'm kind of trying to draw in a big bow | |
28:31 | for you to show you that the equation works . | |
28:33 | The theorem works and it matches with everything we've talked | |
28:36 | about already . So that's all I have of this | |
28:38 | lesson . We're gonna do a lot more problems in | |
28:40 | the next couple of lessons , mostly what I want | |
28:43 | you to understand out of this is conceptually what's happening | |
28:45 | . It is completely possible to add up an infinite | |
28:49 | number of things , but yet get a finite answer | |
28:52 | as a result . That's something that you just have | |
28:54 | to wrestle with and you have to come to accept | |
28:56 | because all of calculus is based on it . You | |
28:58 | will get to that much later when we study calculus | |
29:00 | , but The second half of calculus one is all | |
29:02 | about essentially you're looking at sums of things and I'm | |
29:05 | not going to get into it , but that's what | |
29:07 | we call an integral . And calculus , it's an | |
29:09 | infinite sum of tiny little things . So you have | |
29:12 | to get comfortable with the idea of adding things together | |
29:14 | . So this concept here of this infinite series of | |
29:17 | geometric series giving you an actual finite answer goes directly | |
29:21 | into some of the concepts of calculus , so make | |
29:25 | sure you can understand all of these . I would | |
29:27 | work through this if you can or at least play | |
29:29 | a couple of times to make sure you get it | |
29:30 | , then follow to the next lesson , we're gonna | |
29:32 | start cranking through problems . The actual equation is very | |
29:35 | , very simple . You just have to verify that | |
29:37 | that common ratio is a fraction like we discussed and | |
29:39 | you're good to go to use the equation here . | |
29:41 | So follow me on to the next lesson . We'll | |
29:42 | get more practice right now . |
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