10 - Series and Sigma Summation Notation - Part 1 (Geometric Series & Infinite Series) - Free Educational videos for Students in K-12

10 - Series and Sigma Summation Notation - Part 1 (Geometric Series & Infinite Series) - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is called series and sigma notation . This is part
00:06	one of several lessons now . In the last few
00:09	lessons we have covered in great detail the concept of
00:11	what a sequence is in math . A sequence is
00:14	just the listing of numbers , number comma number ,
00:18	common number like this on and on a sequence of
00:20	numbers . Now we have different kinds of sequences .
00:22	We had arithmetic sequences , we had geometric sequences and
00:26	we have already learned all of that . So if
00:28	you haven't done that material I really need you to
00:30	do that . Before we get into this lesson this
00:32	lesson is about something related . It is called series
00:36	, mathematical series and also the sigma notation which is
00:39	how we write down what a series is in math
00:42	. So before a sequence was just a listing of
00:45	numbers in a series . What we're gonna do is
00:47	add the numbers together . So a lot of students
00:50	look at the concept of a series and they get
00:52	really confused in the beginning . I need to calm
00:54	you down and let you realize that all we're doing
00:56	when we have a series in math is we're taking
00:58	those numbers in the sequence and we're just adding them
01:01	together . That's all there is to it . So
01:03	the concept of a sequence of the concept of a
01:05	series very closely related . Now , before we get
01:08	into the math , I want to give you a
01:09	little bit of motivation . This stuff that we're gonna
01:11	learn here is really getting into what I think the
01:13	upper level of math is before you get into calculus
01:16	and that is calculus is like the crown jewel that
01:19	we use for all advanced math , all engineering calculations
01:23	. You can't get away from calculus . When you
01:25	learn calculus one , the second half of calculus one
01:29	the second half of it is when you study something
01:32	called integration , integration and calculus is based on the
01:36	concept of a series . So when you learn this
01:39	material in this lesson right now , learning the concept
01:42	of a series , it is going to directly feed
01:44	into what you learn for about two months in calculus
01:48	one . It is that important . Okay , and
01:50	then finally I'll just give you a little more motivation
01:52	. Uh this concept of a series , it doesn't
01:54	ever go away . In fact , when you get
01:55	into more advanced physics , when you talk about for
01:57	instance , Einstein's general theory of relativity , which is
02:01	his theory of gravity , all of the equations have
02:05	actually , there's there's there's something called the Einstein summation
02:08	convention which is very closely related to series . In
02:12	other words , there's so many things in gravity theory
02:15	that we have to add together , that Einstein invented
02:17	his own way of writing it down , so that's
02:19	how important it is . The stuff you're learning right
02:21	now feeds in the calculus feeds in the very advanced
02:24	modern physics theories that we use even in the modern
02:28	day . So it's something that's going to be around
02:29	. Okay , so let's go back to the beginning
02:31	. Talk about what the sequences talk about , what
02:33	a series is right down the sigma notation so we
02:36	can solve problems . So let's go back down memory
02:40	lane . Let's talk about what we learned before finite
02:44	sequence . Now , The reason I'm putting the word
02:48	finite in the beginning is because we can actually have
02:51	infinite sequences . You know when you when you don't
02:53	actually have an in point when you just keep on
02:55	adding numbers into the sequence , it would be infinite
02:58	. But here let's just talk about and remember what
03:01	a finite sequences . Let's give an example . A
03:03	finite sequence might be something like three comma seven ,
03:07	comma 11 , comma 15 comma 19 . Now this
03:12	is finite because it has 12345 terms only five terms
03:16	in there . The terms do not go on forever
03:19	. There's just a finite listing of them . And
03:21	it's a sequence . Because there's just comments here right
03:23	now , what kind of sequences this ? Right .
03:26	Because there's there's lots of different kinds of sequences .
03:28	This one's one that we've studied before . We start
03:31	here . If we add four we get the seven
03:34	. If we add four we get the 11 we
03:35	add four and we add four . We can get
03:38	all of these by just adding a number . So
03:40	if you remember back to sequences when we add numbers
03:42	to get the terms that's called an arithmetic sequence .
03:45	So up here under finite sequence , I'm going to
03:48	remind you that this one is arithmetic and it's an
03:53	arithmetic sequence . S E Q . I'm gonna call
03:57	that arithmetic sequence . Now . The concept of a
04:00	sequence is just when you have the numbers listed ,
04:02	when we have a series , we actually are going
04:03	to add those numbers together . So related to this
04:07	finite sequence , which we now know as an arithmetic
04:09	sequence . We have an example of a finite series
04:16	. Now , what do you think this finite series
04:17	is going to be kind of already given the punch
04:19	line away ? That's fine . It's going to be
04:21	three plus seven plus 11 plus 15 plus 19 .
04:27	Now you might look at this and say , well
04:29	that's so easy . All you did was put plus
04:31	signs in between where the comments were . That is
04:33	the only difference between a sequence in a series .
04:36	A sequence is just a listing of numbers in a
04:38	series is just the addition of those numbers . Try
04:41	to burn that in your mind . The reason we
04:43	use series so much in calculus is because most of
04:46	calculus is trying to figure out how to add lots
04:48	of things together . So we have to use the
04:50	concept of series to do that . I don't wanna
04:52	get into calculus now , but that's basically where you're
04:54	skating to , that's where you're going to down the
04:56	road when you learn those advanced things is how to
04:58	add a bunch of things together . So we have
05:00	a series to do that now . Because this series
05:05	is the terms of the series are listed here just
05:08	like the terms of the sequence are here and the
05:10	terms of the series , our arithmetic in nature .
05:12	This thing is called an arithmetic series arithmetic just means
05:22	to get the terms of the series . I just
05:24	AD for AD for AD for AD for because I'm
05:26	adding a constant number to everything . That is how
05:29	I uh that's how I get it . That's how
05:32	I uh label this type of series . Alright ,
05:36	So that's kind of like one little category here .
05:38	Now let's go and talk about something a little more
05:41	interesting . Let's talk about an infinite sequence . Right
05:50	? We haven't really studied infinite sequence too much .
05:52	I mean , we did do a few examples like
05:54	that , but I didn't really make a big deal
05:55	out of it because I knew I was going to
05:56	come to it here . So , let's give an
05:59	example of an infinite sequence . All right . Just
06:02	an example . What about 1/2 comma 1/4 comma 1/8
06:08	comma 1/16 comma dot dot dot . When you see
06:12	the dot dot dots at the end , it means
06:14	the pattern continues and there's no ending to it .
06:16	These sequences and series ended . But this sequence doesn't
06:20	end and that's why it is called an infinite sequence
06:22	. Now , when you look at the terms of
06:24	this infinite sequence , you should see a pattern .
06:26	Notice that this one to go from this term to
06:29	this term . We're not adding numbers , were not
06:31	just adding a constant number like we were doing here
06:33	. This is a little bit more complicated . What
06:34	we're doing over here is this one I'm multiplying by
06:38	a half . If I take this multiplied by a
06:40	half , I'm gonna get 1/4 if I take this
06:42	and multiply by one half , I get an eighth
06:45	if I take this and multiply by one half ,
06:47	I get 1/16 . So because I'm multiplying by common
06:50	number , A common multiplier also called a common ratio
06:53	of terms . We talked about that before . This
06:55	infinite sequence is actually called a geometric sequence . So
07:00	this one is called a geo metric sequence right now
07:07	, what do you think is going to do ?
07:08	If we start adding up these terms , then uh
07:11	we're not going to have an infinite sequence . We
07:12	will have an infinite serious would just take the same
07:19	exact terms , nothing fancy from before . We'll take
07:23	the one half add to it , 1/4 add to
07:27	it , 1/8 add to it . 1/16 plus dot
07:31	dot dot . We never stop adding . So if
07:34	this guy was called an infinite sequence of terms ,
07:37	then this thing is called an infinite series . And
07:39	if this one is called the geometric sequence , this
07:41	one is going to be called a geometric A series
07:48	. All right . So any time you see the
07:51	words series here , series is here . Series is
07:55	here . In your mind . You need to think
07:57	well I'm just adding things up . Any time you
07:59	see the word sequence , you need to think ,
08:00	oh I'm just writing numbers down sequences are important .
08:03	Series are actually even more important . Okay , now
08:07	I know what you might be thinking . How can
08:09	you have an infinite series like this where I'm adding
08:11	and adding and adding and adding like what am I
08:14	gonna do here ? Like Okay , the next term
08:15	after this , if I multiply about half of the
08:17	1/32 then I'll be won over 64 then to be
08:21	won over 1 28 and I'll just keep on going
08:23	and I have more and more and more terms .
08:25	How can I ever add them together ? Like I
08:28	could get my calculator and I can I can keep
08:30	pressing the add button and adding the next term in
08:32	the next term , in the next term . And
08:33	then I could go until the sun explodes , you
08:36	know , billions of years from now until the galaxy
08:38	of the senator . I'm still adding numbers . So
08:41	what is the point of adding up all of these
08:42	numbers ? If I never stop adding , how can
08:44	I possibly do that ? Okay . We're kind of
08:46	getting a little bit touching into the , into the
08:48	boundaries of calculus . I don't want to get into
08:50	it right now , but I want to tell you
08:52	, and this part should blow your mind that sometimes
08:56	even if you add up an infinite number of things
08:59	, you can actually still get a finite number as
09:02	an answer . Now , I'm going to say that
09:04	one more time because if it goes , it goes
09:06	in when you're not the other , you're not gonna
09:08	appreciate it . It's really important . What I'm saying
09:10	is I'm adding up an infinite number of terms but
09:12	notice that each term is getting smaller and smaller and
09:15	smaller . So sometimes and we will discuss later down
09:18	the road how you know when you can do this
09:21	. But sometimes if the terms are decreasing fast enough
09:24	, even though there's an infinite number of them infinite
09:27	on and on and on and forever , I can
09:29	still add them up , quote unquote and get a
09:31	finite number . An actual number . In other words
09:33	, most people think if you keep adding and adding
09:35	and adding , it just goes to infinity . But
09:37	if the terms are dropping really rapidly , I'll explain
09:41	later how we know when that's happening . If the
09:43	terms are dropping fast enough , you can actually get
09:45	an answer a finite number even if you're adding up
09:48	an infinite series like that . And that is actually
09:51	the cornerstone of calculus . We'll talk about that later
09:53	when we get to that subject . But that is
09:55	what we're gonna be doing here when we talk about
09:56	series as well . So you have finite sequences ,
09:59	finite series , that's easy to understand . You add
10:01	things up , you can have infinite sequences an infinite
10:04	series . And even in the case of an infinite
10:05	series , you can still add them up and getting
10:07	number sometimes . So now we have to switch gears
10:11	and say , well it's a real pain to write
10:13	these numbers down with commas and and with plus signs
10:16	. So we have to have a way of doing
10:17	it shorthand , we call it sigma notation Sigma is
10:21	the greek letter greek , capital letter sigma . You're
10:24	gonna be using it over and over and over again
10:26	. So the easiest way to do it is just
10:28	to do it . I'm gonna call it sigma notation
10:34	, I'll tell you right now then it looks really
10:36	scary at first . But actually when I explain it
10:38	it's actually really fun because it's something that makes your
10:42	life easier and it's kind of fun once you get
10:43	the idea of how to do it , what if
10:45	we have the series that looks like this two plus
10:48	four plus six plus eight plus dot dot dot plus
10:53	100 . Now , this is not an infinite series
10:56	. All I've done is put the dots here because
10:58	there's a lot of terms between eight and 100 but
11:00	100 is the last term in this series . Okay
11:03	. It's a series because I have plus signs everyone
11:05	. I'm adding these up and I should ask you
11:07	a bonus question . What kind of series is it
11:09	? Is it an arithmetic series ? Is it a
11:11	geometric series ? Is it ? Neither ? This one
11:14	is in arithmetic series because to get the terms I'm
11:17	just adding to to get four and then to to
11:19	get six I'm adding to to get eight . Every
11:21	term in the series is just adding another number for
11:24	the related sequence . The arithmetic sequence . So this
11:26	is called an arithmetic series and it's a finite series
11:29	because it doesn't go forever . It just stops .
11:33	So obviously writing all these terms down to 100 would
11:36	take a lot of board space . So we want
11:37	to simplify that . Here's how you do it .
11:40	I'm just gonna write it down and we're gonna talk
11:41	about it . You have to get used to join
11:43	this big little E horizontal line down back horizontal line
11:48	. This is how I want you to write your
11:50	sigma's . I don't want you to scribble some lines
11:53	. I don't want you to make an E .
11:54	I don't want you to do a capital anything weird
11:57	. I want it to be horizontal line , diagonal
12:00	line , diagonal line , horizontal line . That is
12:01	the proper way that you write a signal . And
12:03	here is how we're gonna do this . I'll explain
12:05	in a minute . N is equal to one .
12:08	Going up to N is equal to value of 50
12:11	. So in goes from 1 to 50 . And
12:14	what we're doing here inside of the Sigma is two
12:16	times in . Now , if you've ever done any
12:20	computer programming , it's gonna actually a lot easier for
12:22	you to understand this . What I want you to
12:24	see when you look at that sigma is I want
12:27	you to see it as a loop . It's like
12:28	a loop . You go through this little mathematical loop
12:31	. How many times do you do it ? You're
12:33	gonna do it 50 times . Because what is on
12:35	the bottom and what is on the top are the
12:37	boundaries of the loop ? You start you have to
12:41	this variable end is just a placeholder for kind of
12:44	a counter that goes through the loop when in starts
12:47	at one , that's the value of n . You
12:50	go through the loop and then eventually in is two
12:52	and then in his three it's 4567 all the way
12:54	to n is equal to 50 and then the loop
12:56	is stopping , so the numbers on the top and
12:58	the numbers on the bottom , just tell you how
13:00	many times you go through the loop . Now ,
13:02	what do you do in the loop ? What you
13:03	do is what's in here ? This is what you
13:05	do every time you go through the loop . So
13:06	here's how you figure it out , what you say
13:08	is the first time through the loop and is equal
13:10	to one . What I have here is two times
13:12	in so what I'm gonna have is two times and
13:15	is equal to one . That's the first term .
13:17	The sigma means every little term here is added together
13:21	. So the plus signs are automatically included because this
13:23	is a sigma . Then I go through the next
13:26	time through the loop and is now too two times
13:29	in being too is going to be written like this
13:31	and I go back through the loop and then it's
13:33	gonna be in is equal to 32 times three .
13:36	So I write is two times three . I go
13:38	back through the loop and is equal to 42 times
13:41	four , two times four . So you see what
13:43	happens , I go through and through and through and
13:45	through and through and then eventually I get to the
13:46	final value of n , which is the maximum number
13:49	up here , which is 50 and it's two times
13:52	it is equal to 50 dot dot dot Two times
13:56	50 . All right . So what's happening is you're
14:00	going through and you're doing this mathematical calculation as in
14:04	goes bigger , bigger , bigger , bigger now .
14:06	What does this do for us ? What it does
14:08	for us is the following thing . Two times one
14:11	is two , then we have four , Then we
14:14	have six , then we have eight Then we have
14:20	100 . Notice that the series that comes back out
14:23	of this calculation is the same series I started with
14:26	. So what I'm trying to say is that you
14:29	can write the series down with plus times if you
14:31	want to . But what's gonna happen is you might
14:33	have a series with 100 or 200 terms . You
14:36	might have 3000 terms , you might even have infinity
14:39	terms . Because guess what ? Later on ? When
14:41	we have an infinite series , we're gonna use infinite
14:44	infinite terms that need to be added together . So
14:47	we don't want to spend our whole page writing all
14:49	these terms down . So we have a shorthand way
14:51	of putting it . This is a little calculation I'll
14:54	loop that represents the entire series so I don't have
14:59	to write all this stuff down every time . This
15:01	thing right here , this black stuff right here represents
15:04	the series . Because as end goes from 1 to
15:06	50 . When I do this calculation , I recover
15:08	all of these terms that are existing in the original
15:11	thing . So the title of the lesson was called
15:13	series and sigma notation . The first part of the
15:16	lesson is for me to explain what a series is
15:18	, which I've done . The second part is the
15:20	sigma notation . That's this part . So the rest
15:22	of the lesson , all we're gonna do is I'm
15:24	gonna show you how to use the sigma notation .
15:26	We're gonna do a lot of examples . I'll show
15:28	you how to get the terms . I'll show you
15:30	how to take the terms and write down the sigma
15:32	notation . That's all we're doing . You have to
15:34	get comfortable with that because we're going to be using
15:36	that in future lessons . All right , so that
15:40	was our first little example of sigma notation . Let's
15:43	do another one . Here's another series . Here goes
15:46	, my sigma's are not perfect , but that's more
15:48	or less how I want you to write it .
15:49	Let's say the end goes from one up to instead
15:53	of 50 . We're going to only go to up
15:55	to 10 . And let's say the little calculation we're
15:57	doing every time we go through the loop is not
15:59	this , it's going to be in Squared into the
16:02	power of to . So this loop , how many
16:04	times do you think we go through that loop ?
16:06	We only go through this loop 10 times because end
16:09	goes from one up to 10 . In fact ,
16:12	it's probably a good idea for me to label a
16:16	few things for you here . Okay , label a
16:18	few things for you . So this thing is called
16:21	the lower limit of the Sun . This thing right
16:28	here is called the upper limit . Right ? These
16:34	things together are called the limits . These things together
16:37	called the limits of the sum . Okay . This
16:39	thing right here that the actual calculation that we do
16:42	this thing is called the some and some end .
16:48	So when you say some problem says , hey ,
16:50	do it right down the sigma where the sum and
16:52	is equal to whatever that word just means . That's
16:55	the calculation you're doing in there . So we have
16:56	the upper limit , we have the lower limit and
16:58	then specifically this variable that we're using in this case
17:03	, it's in this variable right here . This thing
17:05	is called the index , Right ? So if I
17:09	were to write it in words , I might say
17:11	this series can be written with the variable in as
17:14	the index with a lower limit of one and an
17:17	upper limit of 10 where the sum and is equal
17:20	to end squared . Now , would I ever really
17:22	write that down ? No , but that's what those
17:24	words mean in context . Okay . So how do
17:27	we then proceed ? Okay , what we're gonna have
17:30	is in is gonna be one , then we're gonna
17:32	square it . So it's gonna be one squared then
17:35	we're gonna go through again and it's going to be
17:36	too then we're gonna square it then and it's going
17:39	to be three . Then we're gonna square , you
17:40	see the pattern and it's gonna be four , then
17:42	we're gonna square , we're gonna go all the way
17:43	through dot dot dot to the final value of n
17:47	. Which is 10 . Then we're going to square
17:49	it now . Of course I can square these numbers
17:53	and I can then actually add them up just like
17:55	I could add them up here . I don't actually
17:56	care about finding the sun right now . Right now
18:00	, we're just learning how to write down the sigma
18:02	notation . We're gonna write down how to represent the
18:05	sum in a future lessons . We'll talk about how
18:07	to calculate the sum because there's actually shortcut ways to
18:09	actually Add up . If you had to add up
18:11	all these numbers to 100 , it would take a
18:13	long time you have to use a calculator . There
18:15	are quick ways to add up the numbers . Don't
18:17	worry about finding the sum right now . Right now
18:19	we're just writing down how to represent the sum ,
18:22	right ? So I don't care about squaring the stuff
18:24	and adding it . I just care about you knowing
18:26	that this equals this . One thing I want to
18:30	make absolutely crystal clear is that this index that we're
18:33	using this end ? This variable end , It's what
18:35	we call a dummy variable or a dummy index .
18:38	The actual letter end doesn't matter because notice what's going
18:41	on is I'm just using in to rotate through and
18:43	do this calculation . It doesn't matter what letter I
18:46	use for this , it's just like a like a
18:48	placeholder . So for instance , I can change this
18:51	to the following if I wanted to , I could
18:53	make sigma , right ? I could say variable K
18:57	can go from one up to 10 and then what
19:00	I'm gonna calculate here is gonna be K squared .
19:02	So whatever my indexes , it must be involved in
19:06	the calculation that's going on inside of the some here
19:08	. Otherwise it doesn't make any sense . So I
19:10	have in here and in here and in here and
19:12	in here I have to have K and K .
19:14	I could use a different number of variables I could
19:16	use . JJ is very common . Also , II
19:18	is very common . You can use any number in
19:20	the letter you want . But usually in or I
19:23	or J or K are usually the ones you'll see
19:24	mostly in the books . But if I'm going to
19:26	use this , K , I'm gonna say K is
19:27	one K squared , one squared , then K is
19:30	22 squared , then K is 33 squared . Uh
19:35	, and then plus dot dot dot up to the
19:36	maximum value of k , which is 10 squared .
19:40	Right ? So the , the actual variable that you
19:42	use , whether it's I or J or K or
19:45	in none of it matters . All that matters is
19:46	that you use that variable to loop through the calculation
19:49	. All right Now so far for these examples on
19:54	the board , these are what we call finite series
19:56	because it goes from 1 to 50 . That's a
19:59	finite number of terms . It goes from 1 to
20:01	10 , 1 to 10 . Those are finite number
20:03	of terms . But we already talked about that .
20:05	We can have infinite series , which has an infinite
20:07	number of terms . And so how do we write
20:09	down something like this ? So an infinite series might
20:13	look something like this . Write down my little sigma
20:15	here from K is equal to one . Now ,
20:19	what do you think is gonna happen ? The upper
20:20	limit is not going to be a number . The
20:22	upper limit is going to be an infinity . So
20:24	that's where you put the infinity symbol at the top
20:27	because the upper limit never ends , The maximum term
20:30	, it never comes . It's always out there infinity
20:33	. Never actually happens , it's still going and going
20:35	and going . So if I have K is equal
20:38	to one to infinity , that maybe my terms might
20:40	be something like just as an example to to the
20:42	power of K -1 . Right ? That looks really
20:46	complicated , right ? But what you have to do
20:48	is kind of get rid of that in your mind
20:50	because it really isn't , it isn't really complicated .
20:52	All you're doing is starting at K is one and
20:55	you're putting it in here . So what would these
20:57	terms look like when K is one ? It would
20:59	be 1/2 to the power of K is one .
21:02	Let's write , it is one minus one . We're
21:04	going to add to that looping around when K is
21:06	two , then it will be two to the power
21:10	of two minus one , then it will be one
21:13	over to the power of three minus one , then
21:16	1/2 to the power of four minus one plus .
21:20	And here's where you go dot dot dot because it
21:22	never ever ends . So what you have is K
21:25	is one then Ks to the Ks three than Ks
21:27	four and it never ends . So what does this
21:29	thing really look like on the bottom ? It's gonna
21:31	be one over to to the zero 1/2 to the
21:34	1 , 1/2 to the 2 , 1/2 to the
21:39	three . All I'm doing is doing the subtraction here
21:42	. And what does this come out to to to
21:43	the zeros 1 , 1/1 is one . This is
21:47	going to be 1/2 This , squared is going to
21:51	be 1/4 . This is gonna be 1/2 cubed but
21:54	that's eight . So it's gonna be 1/8 plus dot
21:57	dot dot I forgot right here plus dot dot dot
22:00	as well . So this uh series here can be
22:04	represented as this sigma notation . It's an infinite series
22:09	which means you can see the pattern 1/2 1/4 1/8
22:12	, you know , 1/16 and so on going down
22:15	there basically what's going on is I multiplied by one
22:18	half every time . Uh to get the remaining terms
22:23	which is kind of like what I started with here
22:25	. This is an infinite series here where I just
22:26	started with one half of one half , 1/4 181
22:29	16 . This is the same series except the first
22:31	term is not one half . The first term is
22:33	one so 11 and one half and 1/4 and 1/8
22:37	and 1/16 and one 32nd and so on , it's
22:39	going to keep going . So instead of writing all
22:41	these things out , I just write the sigma notation
22:44	. Now . There's two things to skills here .
22:47	First skill is looking at the sigma notation and writing
22:51	down the series . That's what we're focusing on now
22:53	. Later on we're gonna reverse it where I will
22:55	give you the series and you have to come up
22:58	with the sigma notation right now , before we get
23:00	there , I want to make one other thing .
23:03	It's important for you to know is that here I
23:06	have K . Going from one to infinity . And
23:08	here are the terms the some end that yields this
23:12	thing . There's more than one way usually to write
23:15	the summation notation . Um to give you the same
23:19	series for instance , let me go down here and
23:21	drop down below . Just explore this with me a
23:25	little bit . What if instead of this , what
23:27	if we get the summation of K0 up to infinity
23:32	. Notice I'm not going from one to infinity .
23:33	I changed the limits and I go from zero to
23:35	infinity of one over to to the K . Notice
23:40	that this is really similar to this . What's going
23:44	on here is this is in the expo and I
23:45	have a k minus one , but I changed the
23:48	calculation so it doesn't say k minus one . What
23:50	happens is I add one in the expanded up here
23:53	because I've advanced one in the exponents . I compensate
23:56	for that by subtracting one in the index . So
23:59	if I mess around with the sum and with the
24:01	calculation there and change the expanded or change something related
24:05	to the index , the variable . Then if I
24:07	change anything regarding that , I also need to go
24:10	adjust my limits of summation to make sure I get
24:13	the same term . So if we go through this
24:14	calculation , we can see that this will be 1/2
24:18	to the power of 0 , 1/2 to the power
24:20	of one . When K becomes 1 , 1/2 squared
24:24	one over to third plus dot dot dot . And
24:27	you can see these terms of the same terms as
24:29	these , which are the same terms as these .
24:30	So this series can be represented by this or this
24:36	series could also be represented by this and I could
24:40	if I wanted to change this even further and then
24:42	change the limits even more drastically . But these are
24:44	the same . In other words , it's not really
24:48	more correct to say this one is the right answer
24:51	or this one is the right answer . As long
24:52	as those two sums the summations , the summation convention
24:55	, as long as it's giving you the same terms
24:58	, then they're both equally correct . They represent the
25:00	same things . But still sometimes when we do problems
25:02	, one of them looks a little simpler than others
25:04	, you know , like 5/10 and one half They
25:08	both represent half the pizza , right ? 5/10 and
25:11	one half they represent the same thing . But we
25:13	usually say one half is simpler . So that's what
25:15	we usually like this one here is it looks simpler
25:18	. So usually will probably write something like this down
25:21	, although this gives exactly the same terms . So
25:23	just keep that in mind when you get your answers
25:25	to these problems and you look at my answers or
25:27	you look at answers in your textbook or whatever .
25:29	As long as it could be a little different if
25:31	you adjust the exponent here , if you're just anything
25:34	related to K adding or subtracting and you compensate by
25:38	messing around with the limits of summation , then it
25:41	can still be correct . That's all I'm trying to
25:43	tell you . Alright , so finally let's do a
25:45	couple of quick problems . None of these are hard
25:47	, but let's get some examples . What if I
25:50	do the limits from N is equal to one up
25:53	to six of n plus 10 . What I want
25:57	you to do is I want you to write it
25:59	in expanded form . I want you to take the
26:00	some the summation notation and I want to expand this
26:04	into the full blown some what do we do ?
26:08	We just go through as a loop . Like a
26:09	little computer program when it is one end goes into
26:12	one here , one plus 10 . Now don't do
26:14	too many things at once . Don't try to add
26:16	one plus 10 in this step . I don't want
26:18	that . I want you to just put the number
26:19	in there because if you do too many things at
26:21	once , you will make a mistake when the problems
26:23	get harder . What I want you to write down
26:25	for the first term is in being 11 plus 10
26:28	. This is the first term . When I put
26:30	it in here , I'm going to add to that
26:32	because it's a summation in being too there will be
26:35	two plus 10 , right ? Then it will be
26:38	three plus 10 , Then it will be four plus
26:42	10 . Right ? We'll go down here , then
26:44	it will be five plus 10 , Then it will
26:47	be six plus 10 . Because then finally in has
26:52	gotten to its maximum value of six . Now ,
26:54	in the next step , then you go and add
26:56	you say this is 11 . This is 12 ,
26:59	13 , 14 , 15 , 16 . Now ,
27:04	I know that you can all gravity calculator and add
27:06	these up . I don't want you to do that
27:07	. What I want you to do is circle this
27:09	because this is what we call expanded form form ,
27:16	right ? All I care about is that you can
27:18	translate what the summation is telling you the sigma and
27:21	write down the some . That's all I care about
27:23	. I know that you can all grab the calculator
27:24	and add those together . Isn't that isn't helpful to
27:27	me what I want you to know what I want
27:29	to know is if you can take the some the
27:30	summation sigma and blow out the terms later on .
27:34	And a few lessons , we'll talk about how to
27:36	add these things together . I mean , obviously you
27:37	can have these . But what happens if there's 1000
27:40	terms ? How do I add them up without spending
27:42	my whole life doing it ? All right . So
27:45	that's writing an expanded form . We're gonna do one
27:47	more like that and we'll call it a day and
27:48	wrap up this lesson . What if I have The
27:51	summation end is one up to six and the sum
27:55	and is 2 to the power of n . Okay
27:59	, So what you do here is you say In
28:02	his one , so two to the power of one
28:05	. Then in goes to to to to the power
28:07	of to then in goes to 32 to the power
28:10	of three . And you see the pattern here too
28:11	. To the power of four , two to the
28:14	power of 52 to the power of six . I
28:16	stopped at six because N has finally reached its maximum
28:19	number which is six and I stop . If there
28:21	was an infinity here , then I would just put
28:23	dot dot dot and I would keep the terms would
28:25	go forever . But this is a finite series .
28:28	All right . And so then you all know that
28:30	this is to this is for this is eight .
28:33	This is 16 . This is 32 and this is
28:36	64 90 the calculator . To figure these last terms
28:40	out . But you can see what's happening is it's
28:41	doubling , doubling and doubling again . This is what
28:44	we call expanded form . That's what I want you
28:46	to write down . All right . So , in
28:48	this lesson , we have learned a tremendous amount of
28:50	material . It's probably one of the more important lessons
28:52	uh in recent times because it has such far reaching
28:56	implications to calculus and other events math . Beyond that
28:59	. We started out by saying that we have the
29:01	concept of a sequence . It's just a listing of
29:03	numbers . When you add up the terms of the
29:05	sequence , we call it a series . It's a
29:07	finite series when it has a limited finite number of
29:10	terms . Right . And then we said , we
29:12	can have an infinite sequence where the terms of the
29:15	sequence just keep going and going and following the same
29:17	pattern forever separated by commas . If we add up
29:20	those terms of the infinite sequence , we arrive at
29:23	what we call an infinite series . Now , because
29:25	the terms of the sequence can be arithmetic or geometric
29:29	. Then we also have instead of just arithmetic sequence
29:31	in geometric sequence . We also now have arithmetic series
29:35	and geometric series . There are other kinds of series
29:38	in math that you'll learn in more advanced classes .
29:40	It's not limited to just these . There's tons of
29:42	other kinds of series . There's power series , there's
29:45	all kinds of things . I don't want to get
29:46	into them . But there are other series that have
29:49	other names that will learn down the road . Then
29:51	we talked about the summation notation . It's basically a
29:54	loop that you just crank through calculating the answers uh
29:58	, through the some end and you just follow the
30:00	limits of summation here . Uh and basically that's it
30:04	. You just crank through it and then we have
30:06	the idea that you can represent an infinite series .
30:08	Also with the summation notation , where infinity becomes the
30:11	upper limit . Importantly , we also said that if
30:14	you change , add or subtract numbers to the index
30:18	in here , you can compensate for it by changing
30:21	the limits of of the summation so that you can
30:24	arrive at what looks like different answers , but they
30:26	give exactly the same series . So they're both basically
30:30	correct . Uh and that's going to pop up in
30:32	problems later down the road . And then we just
30:34	did a couple of problems explaining how to write an
30:36	expanded form . Very soon . I will give you
30:39	the expanded form and ask you to write the summation
30:42	notation for me . So what I want you to
30:44	do is solve every one of these . Make sure
30:46	you understand it all . Follow me on to the
30:48	next lesson . We have several lessons in seed and
30:50	series and sigma notation and we'll get into arithmetic series
30:54	, geometric series , a lot more detail , infinite
30:56	series as well as we progress through the lessons here
30:58	in series and sigma notation .

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