The last thing I want to talk about is summation and product notation.
So again your class might not have done this. So on Friday, I…
So at the end of the week… the Friday of the week…
I gave my class a bunch of sample problems to go through and to solve
using all the techniques up to this point. So we've done...
by Friday we will actually been 12 lectures - well 11 lectures completed.
So you have a lot of proof techniques, and now when you actually read a problem,
it's tough to know where to start the problem.
So that was part of what I did on Friday,
and the other part of what I did was summation/product notation.
Now why did I do this? So next week, we're
heading into something called Mathematical Induction,
and this notation will appear everywhere.
So my theory is that you should see this.
Spend some time with it over the weekend,
actually make sure you understand it because again
if you're going to get problems that involve this notation
and you have to think about what the notation means,
it's going to be harder to solve the problem.
So spend some time with this notation this week and
really make sure you understand what it means so that
when you see problems with it, you know what to do.
You know what it means you don't have to think about what these words mean, okay?
It's really important.
So okay, summation and product notation.
Well let a 1 to a n be a sequence of real numbers.
In this case n real numbers.
We're going to write the summation from i equals 1 to n of a i
is going to be defined to be: “a1 plus a2 all the way up to a n.”
We'll see some examples in the next page. But basically what does this mean?
It means add all of the elements
a sub i from i equals 1 to n.
So if you understand programming language stuff, this is similar to a ‘for loop’
something like this. A ‘for loop’ of a sum.
We call i the index variable. We say that 1 is
the starting number and n is the upper bound or the finishing number, the final index,
all of these words, okay?
We can also write summation notation as follows.
So we have sets, we can write x is an element of S of x
to mean the sum of the elements of S.
Of course this has to make sense, so S has to have some addition operation on it.
So let's say S is the set of integers, or real numbers, something like this.
If S is the set of all hockey players on a hockey team,
then clearly this doesn't make any sense, so it has to make sense to actually do this.
Similarly, we define product notation.
So the product from i equals 1 to n of a i.
We're going to define that to be the product from a1, a2, all the way up to a n.
So this is a big product.
So here I should note also I've used for the first time
“colon equals”.
“Colon equals” means “defined to be”, okay?
So I’m defining this notation to be this product.
I'm defining this notation to mean this sum.
I'm defining this product notation over x is in S
this is the product of elements in S, okay?
This product notation is a capital letter pi by the way.
Slash prod in LaTex.
Slash sum is the big summation. Sigma.
Capital sigma that's what it is.
We also make a couple of following conventions: so if
the starting index is bigger than the final index,
we're going to call the sum the “empty sum”.
We're also going to define the sum over the empty set to just be 0.
That's just what we're defining it as,
it's just a definition it makes life easier,
it works in a lot of nice situations.
Take it as it is, okay?
We're also going to define the product if
the starting index is bigger than the final index to be 1.
And again this is the same as the product over the empty set.
Just notation, just to make life easier, okay,
the empty product is 1, the empty sum is 0.
Just makes life easier.
You'll see if you actually read papers and stuff that use this stuff.
It just usually works out in this case.
Some examples, so we're going to finish off with a couple examples.
Make sure we understand the notation.
So the sum from i equals 1 to 4 of i squared, what does that mean?
Well how do we do this? We start off with i equals 1,
we plug it in, we square it. That's the first term.
Increment your counter to 2. 2 is less than or equal to 4, plug in 2.
2 squared.
Increment the counter to 3. 3 is less than or equal to 4,
plug in 3, 3 squared.
Increment the counter you get to 4. 4 is less than or equal to 4,
plug in that number 4. 4 squared
into the i squared part.
Increment the counter we get to 5. 5 is bigger than 4, we stop.
Those are all the terms.
Then we can just add them up and we get 30.
Similar with product notation,
right, same sort of idea. Plug in i equals 1,
we're going to get i squared, so 1 squared,
and we're going to multiply that by the next term,
right, so instead of 1, we plug in 2.
So 1 goes to 2, 2 we plug in 2 squared.
Increment the counter to 3, 3 we plug in 3 squared.
Write that down. Again, it's a product.
Increment the counter to 4. 4 is less than or equal to 4, plug in 4, 4 squared.
Increment the counter we get to 5. 5 is greater than 4, we stop.
Take the product of all these numbers: 1 times 4, times 9, times 16,
so 64 times 9 is 576.
Another example, you won't see this too often:
i going from 1 to 3 and a half of i.
You go 1, 2, 3, and 4 is bigger than 3 and a half so you're done.
Usually you write this with integers, you can do it with decimal numbers too.
I've actually never really seen this, but this is one way to do it…
define it.
Here's more of a convoluted example: so for
k, a natural number fixed, what if we wanted the sum from
i equals k to 2k of 1 over k?
Well again, it's the same idea. Start with k, plug it in,
then increment the counter to k plus 1.
If k plus 1 is less than 2k, we can write it, and you keep going.
k plus 1, k plus 2, k plus 3, we go all the way up to… what's the last term?
1 over 2k.
Now something to think about, right? If we're going to keep doing this,
k, k plus 1, k plus 2, k plus 3,
what's eventually going to happen is we're going to write k in two different ways.
We're going to write it as k plus some number,
k plus a million who knows.
Whatever… it depends on what k actually is.
so let's say k plus j,
and we're going to also write it as 2k, right?
Eventually k plus j is going to equal 2k for some j.
So don't let that confuse you here. Again it's just you start from the index
1 over k, and you just increment all the way up to 1 over 2k.
That's all this means.
There are other convoluted ways to write it, but
this is the main idea, okay, and you can write it…
so in these upper cases,
it was easy because we knew what the index was.
If you don't know what the index is, you can use dots.
The summation notation is supposed to take the place of these dots,
and makes it look a little cleaner and nicer,
but if you need to when you're learning this notation,
feel free to write down the dots if it helps you out.
So that's all I have to say for this video.
Again hopefully this helped you, gave you some ideas,
clarified things from class. If not, again,
as always, send me an email, send me a post on Piazza or wherever.
I'm usually around I'll hopefully answer your questions.
Really I just hope this helped.
I'm trying to make this a good learning tool,
so let me know. Any feedback is great,
and thank you very much and good luck.