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The last thing I want to talk about is 
summation and product notation.

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So again your class might not 
have done this. So on Friday, I…

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So at the end of the week… 
the Friday of the week… 

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I gave my class a bunch of sample 
problems to go through and to solve

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using all the techniques up 
to this point. So we've done...

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by Friday we will actually been 12 
lectures - well 11 lectures completed.

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So you have a lot of proof techniques, and 
now when you actually read a problem,

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it's tough to know where 
to start the problem.

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So that was part of 
what I did on Friday,

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and the other part of what I did 
was summation/product notation.

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Now why did I do this?
So next week, we're

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heading into something called 
Mathematical Induction,

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and this notation will 
appear everywhere.

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So my theory is that 
you should see this.

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Spend some time with 
it over the weekend,

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actually make sure you 
understand it because again

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if you're going to get problems 
that involve this notation

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and you have to think about 
what the notation means,

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it's going to be harder 
to solve the problem.

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So spend some time with 
this notation this week and

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really make sure you understand 
what it means so that

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when you see problems with 
it, you know what to do.

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You know what it means you don't have to 
think about what these words mean, okay?

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It's really 
important. 

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So okay, summation
and product notation.

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Well let a 1 to a n be a 
sequence of real numbers.

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In this case n 
real numbers.

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We're going to write the summation 
from i equals 1 to n of a i

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is going to be defined to be: “a1 
plus a2 all the way up to a n.”

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We'll see some examples in the next page. 
But basically what does this mean?

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It means add all 
of the elements

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a sub i from i 
equals 1 to n.

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So if you understand programming 
language stuff, this is similar to a ‘for loop’

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something like this. A
‘for loop’ of a sum.

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We call i the index 
variable. We say that 1 is

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the starting number and n is the upper bound 
or the finishing number, the final index, 

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all of these 
words, okay?

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We can also write summation 
notation as follows.

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So we have sets, we can write 
x is an element of S of x

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to mean the sum 
of the elements of S.

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Of course this has to make sense, so S 
has to have some addition operation on it.

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So let's say S is the set of integers, 
or real numbers, something like this.

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If S is the set of all hockey
players on a hockey team,

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then clearly this doesn't make any sense, 
so it has to make sense to actually do this.

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Similarly, we define 
product notation.

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So the product from 
i equals 1 to n of a i.

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We're going to define that to be the 
product from a1, a2, all the way up to a n.

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So this is a 
big product.

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So here I should note also 
I've used for the first time

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“colon equals”.

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“Colon equals” means 
“defined to be”, okay?

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So I’m defining this notation
to be this product.

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I'm defining this notation 
to mean this sum.

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I'm defining this product 
notation over x is in S

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this is the product of 
elements in S, okay?

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This product notation is a 
capital letter pi by the way. 

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Slash prod 
in LaTex.

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Slash sum is the 
big summation. Sigma.

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Capital sigma
that's what it is.

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We also make a couple of 
following conventions: so if

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the starting index is 
bigger than the final index,

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we're going to call the 
sum the “empty sum”.

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We're also going to define the sum 
over the empty set to just be 0.

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That's just what 
we're defining it as,

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it's just a definition 
it makes life easier,

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it works in a lot 
of nice situations.

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Take it as it is, okay?

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We're also going to 
define the product if

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the starting index is bigger 
than the final index to be 1.

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And again this is the same as 
the product over the empty set.

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Just notation, just to 
make life easier, okay,

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the empty product is 1, 
the empty sum is 0.

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Just makes life easier. 

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You'll see if you actually read 
papers and stuff that use this stuff.

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It just usually works 
out in this case.

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Some examples, so we're going to 
finish off with a couple examples.

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Make sure we 
understand the notation.

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So the sum from i equals 1 to 4 of 
i squared, what does that mean?

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Well how do we do this? 
We start off with i equals 1,

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we plug it in, we square 
it. That's the first term.

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Increment your counter to 2. 2 is 
less than or equal to 4, plug in 2.

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2 squared.

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Increment the counter to 3. 
3 is less than or equal to 4,

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plug in 3, 
3 squared.

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Increment the counter you get 
to 4. 4 is less than or equal to 4,

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plug in that number
4. 4 squared

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into the i 
squared part.

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Increment the counter we get to 
5. 5 is bigger than 4, we stop.

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Those are all
the terms.

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Then we can just add 
them up and we get 30.

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Similar with 
product notation,

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right, same sort of idea. 
Plug in i equals 1,

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we're going to get i
squared, so 1 squared,

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and we're going to multiply 
that by the next term,

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right, so instead 
of 1, we plug in 2.

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So 1 goes to 2, 2 we 
plug in 2 squared.

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Increment the counter to 
3, 3 we plug in 3 squared.

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Write that down. 
Again, it's a product. 

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Increment the counter to 4. 4 is less 
than or equal to 4, plug in 4, 4 squared.

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Increment the counter we get to 
5. 5 is greater than 4, we stop.

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Take the product of all these
numbers: 1 times 4, times 9, times 16,

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so 64 times 
9 is 576.

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Another example, you
won't see this too often:

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i going from 1 to 
3 and a half of i.

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You go 1, 2, 3, and 4 is bigger 
than 3 and a half so you're done.

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Usually you write this with integers, you
can do it with decimal numbers too.

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I've actually never really seen 
this, but this is one way to do it…

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define it.

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Here's more of a 
convoluted example: so for

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k, a natural number fixed, what 
if we wanted the sum from

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i equals k to 
2k of 1 over k?

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Well again, it's the same 
idea. Start with k, plug it in,

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then increment the 
counter to k plus 1.

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If k plus 1 is less than 2k, we 
can write it, and you keep going.

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k plus 1, k plus 2, k plus 3, we go all 
the way up to… what's the last term? 

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1 over 2k.

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Now something to think about, right? 
If we're going to keep doing this,

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k, k plus 1, k 
plus 2, k plus 3,

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what's eventually going to happen is
we're going to write k in two different ways.

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We're going to write it 
as k plus some number,

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k plus a million 
who knows.

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Whatever… it depends 
on what k actually is.

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so let's say 
k plus j,

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and we're going to
also write it as 2k, right?

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Eventually k plus j is going 
to equal 2k for some j.

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So don't let that confuse you here.
Again it's just you start from the index

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1 over k, and you just increment 
all the way up to 1 over 2k.

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That's all 
this means.

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There are other convoluted
ways to write it, but

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this is the main idea, okay, 
and you can write it…

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so in these 
upper cases,

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it was easy because we 
knew what the index was.

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If you don't know what the
index is, you can use dots.

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The summation notation is supposed 
to take the place of these dots,

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and makes it look a little
cleaner and nicer,

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but if you need to when 
you're learning this notation,

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feel free to write down the 
dots if it helps you out.

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So that's all I have 
to say for this video.

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Again hopefully this helped 
you, gave you some ideas,

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clarified things from 
class. If not, again,

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as always, send me an email, send 
me a post on Piazza or wherever. 

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I'm usually around I'll 
hopefully answer your questions.

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Really I just 
hope this helped.

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I'm trying to make 
this a good learning tool,

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so let me know. Any 
feedback is great,

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and thank you very 
much and good luck.