 Welcome back to our lecture series Math 1210 Calculus 1 for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Missildine. We are starting a new unit, sort of the final unit for Calculus 1 here about integration. We actually saw a little bit of that last time as we talked about indefinite integrals aka anti-derivatives. Chapter 5 is going to start developing the ideas behind what's known as a definite integral and show that the definite integrals we talk about in this chapter are related to the indefinite integrals we talked about at the end of the last chapter. Now, prior to jumping into what an integral is, a definite integral, it's actually worth our time to develop the notation of summation for functions that is adding things together, or sometimes called sigma notation. Sigma here is going to stand for some. The idea is this, if we have a sequence of numbers, so we just have the list of numbers, a1, a2, a3, a4, a5, etc. This could be any sequence, it could be just the counting sequence, 1, 2, 3, 4, 5. It could be the sequence of primes, 2, 3, 5, 7, 11. It could be the Fibonacci sequence, 1, 1, 2, 3, 5, 8, whatever. We just have a sequence of numbers and we want to talk about adding those numbers together. So when you see an expression like this right here, this sigma notation, it looks like this Greek letter sigma, which you might see around a college campus, maybe for a fraternity or sorority or something like that. Sigma, it kind of looks like an E, but this symbol right here, sigma is essentially the Greek alphabet equivalent of a capital C and that C is going to stand for sum because we're adding together things here. And so let's talk about the anatomy of this sigma notation here. So whenever you see the sigma, that means we're going to add some stuff together, we get a sum. What are we adding together? Where we're going to add together this sequence, a sub i, these are summands, the things that get added together, or sometimes we call these the terms of the sum. All right. Here at the bottom, you're going to see a character right here, it's an i here. This is going to be a variable, it's often called the index, hence why we used an i right there. And this thing is going to act like a dummy variable. It's going to keep track of where you are in the sequence. It's going to represent a generic placeholder of what could be inside of the sum right here. If I was to use a computer science analogy here, this letter i is like the variable of like a for loop, if that makes any sense to you. And so this number right here, m, this is going to be our initial value. This is the number that i starts at. And then this number on top, you can hardly see it anymore, this number on top is going to represent our terminal value. That is, it's the value that it stops at. So our sum will range from m to n. Let me get this out of the way. No, certainly pause the video right now if you need to. Oh, terminal value. What the junk does that even mean? Try that again. Terminal value there. And so if I get this out of the way for a sec, we have this sum of the terms a i, where i goes from m to n. And so you see that there's a starting term, m and m. So m represents where it starts and n represents where it ends. And the i just kind of keeps track of where you are along the way. And so this is going to be the sum that goes from a m all the way up to a n. Let's look at some specific examples to see what's going on here. So if you see a sum, so sigma here means sum. So if you take the sum, just look in this first one, of course, we're going to take the sum where i ranges from 1 to 4 and we add up i squared. So if we were to break this up into pieces, what this means is we're going to take 1 squared plus 2 squared plus 3 squared plus one of my favorite games as a kid, 4 squared. And so we go from 1 to 4 in this sum. And we range over all the possibilities of i. And so we just take together the sum of squares. And so we can simplify the sum, of course, because 1 squared is 1, 2 squared is 4, 3 squared is 9, 4 squared is awesome. So we get this. And so let's see 1 and 9 makes 10, 4 and 16 makes 20. So this sum will add up to be 30 right here. But mostly what I care about right now is we understand that this sigma notation represents this expansion right here. For the next one, we're going to take the sum of i as i goes from 1 up to n. And so we're going to get 1 plus 2 plus 3 plus 4 plus 5. How long does this keep on going? We're supposed to go up to n. Well, n's a number. It's a variable I haven't specified yet. So sometimes we have to do dot, dot, dot to let you know how far we're going. We're going to go up to n right here. So we get 1 plus 2 plus 3 plus 4 plus 5 plus dot, dot, dot up to n here. And so these ellipses that you see between the plus, the two pluses here, these ellipses are just representing that the pattern will continue on in the, the pattern will continue using the trend that's already been established here. And sometimes these dot, dot, dots can be extremely ambiguous, absolutely ambiguous. And the purpose of the sigma notation is to try to remove the ambiguity. So we know exactly what we mean by the pattern because one can sometimes get trapped in a pattern and not know what the next step is. So I mean, for example, if I give you the pattern 1, 2, dot, dot, dot, it's like, what's the next number? Is it like 1, 2, 3, 4, 5, right? Is it just county numbers? Maybe just increment, it just increases by 1 each time. Maybe it's like 1, 2, 4, 8, 16, 32. Maybe it like doubles each time, or who knows? We might have like some type of Fibonacci growth going on here. The issue is when you don't offer enough terms, it's too difficult to know what the pattern's going to be. So it's always a good idea when you have a pattern that you list a lot of terms so that the reader can know exactly what you're talking about. But it honestly, no matter how many terms you list, there's always a level of ambiguity whatsoever. So it's important to have a general form like right here. We just represent a function that represents the next term in the sequence. Now, one thing that's important to notice when it comes to these sigma notations, the index or the dummy variable, whatever you want to call it, it doesn't really matter. We can use the symbol i, we can use the symbol j, we can use the symbol k. This is some of the most common symbols used for the dummy variable here. But the index, it's just a letter. It doesn't matter what it is as long as it's not sort of overloaded in its meaning later on. Also, there's nothing that requires our sequence or sum here to start at 1. If we want to start counting at 0, if that's appropriate, we could do that. If we want to start counting at 12, we could take the sum of the 12th term up to the 20th term. If that's appropriate, we can do that. And this notation is built for that type of thing. So if we take the sum, where j ranges from 0 to 5 of this sequence, we would take 2 to the 0 plus 2 to the 1 plus 2 to the square plus 2 3rd to the 3, 2 to the 4th, and 2 to the 5th. And so we can compute each of these. 1, 2, 4, 8, you probably know your powers at 2 if you like to play the game 2048, assuming you're any good at the game. So we get 1 plus 2 plus 4 plus 8 plus 16 plus 32. That kind of looks like a 3 squared, 32 there. If we add those all up, that adds up to be 63. And again, the specific number is not of much interest to us at the moment. It's about understanding what does this sigma notation even mean. And so for this last one right here, we're going to take the sum of our k ranges from 1 to n. Again, the dummy variable doesn't really matter. We end up with 1 plus a half plus a 3rd plus a 4th. You don't have to write out every single term under the sum, but just enough so that's clear to everyone what exactly is the pattern. The last term in this sum is going to be 1 over n, which is this terminal value on the top right there. Can we go the other way around? Can we find a compact sigma to represent the sum of a big sum of some kind, right? So we have this one right here, 2 cubed plus 3 cubed plus dot dot dot up to n cubed. We're supposed to figure out what the pattern means at this moment. Well, I see a bunch of cubes that are adding together, so a very natural candidate would be something like this. We take sigma where i ranges from what's the first term in the sum. Well, that's going to look like a 2. Then we go up to the biggest one, n, and we're adding up together i cubed. So that's a possibility. That gives us a sum of these things, but it turns out this is not a unique way. There's more than one way you can actually represent the sum. Another way is, I'm going to use a different variable. So again, if you change the name of the variable, that doesn't make much of a difference. But if you take k equals 1 and you want to go up to n minus 1, how does that work? Well, you could take k plus 1 cubed. And if you think about how that expands, right? If you take k equals 1, you're going to get 1 plus 1, which is cubed. That's a 2 cubed. Then you're going to get 2 plus 1, 3 cubed. And then this will proceed all the way up to n minus 1 plus 1 cubed. If we simplify this thing, we get back the original sum that we were looking for. This second example right here is what's known as a index shift. This would be like the same thing as performing a horizontal shift to a graph. We can lower the initial value and also the terminal value. We can lower the range of this sum by one, but then each variable would have to increase by one. It's the idea, if you want to move left, you got to add one to the function. If you want to move right, you have to subtract one from the variable. You have to turn right to go left. It was just as confusing for him as well. So try coming up with some sums on your own and write them up as a sigma notation. We'll talk some more about the sigmas next time, some properties we can use to compute them more simply. See you then.