Infinite Geometric Series & Intro to Limits in Calculus - Part 1 - [18] - Free Educational videos for Students in K-12 | Lumos Learning

Infinite Geometric Series & Intro to Limits in Calculus - Part 1 - [18] - Free Educational videos for Students in k-12


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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