Hello everyone. So in this video, I'd like to talk about summations and products,
but first I'd like to review what the notation means. So
recall that when we have a sum from i equals 1 to n of a i,
this is literally the terms
when you...you start by plugging in i equals 1,
you get the term a1,
then you increment i so you go to i equals 2.
If 2 is less than or equal to n, you plug it into here and you repeat
until you get to the very end, so a1 plus a2 all the way up to a n.
Again if n, for example, was 1,
there will only be one term in the sum.
If we change this sum to the product, the only thing that changes is the operation between them,
so instead of sums, they're products now.
With this in mind, let's answer the following questions,
so question 1: "Write the following summation using a summation starting from 1."
So here the summation starts at 3 goes up to 28,
and you have this expression here.
So how do we change the bottom index
to start from 0? Well the idea
is
you're going to use
a new letter. So we're going to define a new letter, let's call it j,
and we're going to define it to be i minus 2.
Now you can rewrite the summation using the same letter but I'm going to use a different letter just
for ease of notation.
So now that j is i minus 2, when i is 3, j is 1
and when i is 28 j is [26] so...
if I replace i
with j plus 2
and write up the summations, let's see what we get.
Let's see what we get. So when we plug it into the summation,
i equals 3 becomes j equals 1,
28 becomes 26 that's [28] minus 2,
the i becomes j plus 2, and the i plus 1 becomes j plus 2 plus 1,
so let's write that as j plus 3. Let's make it
in its simplest form.
And these two summations are equal. Right?
This is what you call substitution. You could substitute a letter for a different letter,
and you can modify your summation
accordingly.
Now, for example,
you can also say that these two things are equal. We can also just
expand some of the terms out which might be a good exercise so
I'm just... let's call it a "for fun" section.
So the sum
is going to be 3 times 3 plus 1
to the power of 3.
So if we start to evaluate this sum, right, the first term is 3 times 3 plus 1 cubed, right,
I'm just plugging in 3. I increment 3 and I go to 4,
and I get the next term:
the 4 times 4 plus 1 cubed.
Now you increment i to go to 5, 5 is less than 28,
and I continue and I continue and I continue.
When's the last term? Well the last term is going to occur when i equals 28 so I'm going to get 28,
28 plus 1 cubed.
So hopefully we all agree that that's true
and then we can do this with the other sum as well.
Again, this is just for practice; just to make sure we understand
the notation and that we can actually do this.
So if I see the other sum,
I'm going to plug in the first term which is 1 plus 2,
1 plus 3 cubed,
plus 2 plus 2, 2 plus 3
cubed,
and then the last term, again 3, 2, 3…
so I plug in 1, I plug in 2, I plug in 3, and I go all the way up to 26.
So let's see what we have, sorry that was a typo,
so we plug in the 1 term into here so 1 plus 2, 1 plus 3.
Hopefully we agree that those two things are equal,
then we have 2 plus 2, and 2 plus 3
so 4 and 5, those two things are equal, and then we keep going
and at every step, even though I didn't show all the steps,
28 is 26 plus 2, 28 plus 1 is the same as 26 plus 3.
These two sums are actually equal, every term is the same.
Okay?
Hopefully this gives you a idea of how to change the indices so that
you can start it at whatever number you'd like,
and this might be important for certain applications.
Question:
"Write the sum from i equals k plus 2 to 3k plus 1 i
as a summation from i equals k to i equals 3k
of i."
You want to use other terms as well, right, because there's no way these two things are going to be equal.
So let's see that now.
Let's look at this, so
I'm not going to expand this sum out like I did the last one, just for
sanity's sake.
We should be able to do this just by looking at it, right? Where do these differ? Well…
let's look at this equality.
Now the terms here, there's a couple of extra terms. Right? We start at k,
but this sum started at k plus 2, right? So this one's going to have a k,
a k plus 1, a k plus 2, k plus 3 all the way up to 3k, right,
but the k and the k plus 1 terms aren't the same.
So if I'm going to include the k and k plus 1 terms in the sum,
I'd better remove them.
Right now, if you expand this out, this is a sum from k, k plus 1,
k plus 2, k plus 3, but now I’m subtracting off k and k plus 1.
So I could condense this to a sum starting at k plus 2,
so these two things at the bottom end are equal.
Now on the top end though, this one has a term 3k plus 1 and this one stops at 3k.
So to account for that, I need to include the 3k plus 1 term. If I plug in
3k plus 1 in for i, I'm just going to get 3k plus 1.
You could imagine changing this question slightly.
For example, if I made everything squared,
instead of i I had i squared, then I would instead of just
k and k plus 1,
I would have squares everywhere.
So let me do the changed question.
Sorry, I didn't actually put up the answer to the other one, sorry. It didn't render.
So what if I changed it to, "Write i squared as a summation",
so write this sum as a summation like this with other terms.
Here again, I'm starting from i equals k to 3k of i squared.
The kth and the k plus first terms,
these are
extra on this side,
right, this starts at k plus 2 but this starts at k, so I need to
remove those in order to get the equality on the bottom end of these two sums to be equal.
And on the top end, well this one
finishes at 3k plus one and this one finishes at 3k
so I need to add in that extra term that I've lost.
Now, for example, if I change the 3k plus 1 to a 3k plus 2,
then I would just simply add the extra 3k plus 2 term.
And there you have it.
These are all ways to manipulate summations,
these aren't all of the ways but these are some of the ways to do so.
So hopefully this gives you a better understanding of the notation
and helps you out for your midterm. Best of luck.