Topic E: Lesson 29: Geometric series
What I want to do in this video is introduce you to the idea of Sigma notation, which will be used extensively through your mathematical career. So let's just say you wanted to find a sum of some terms, and these terms have a pattern. So let's say you want to find the sum of the first 10 numbers. So you could say 1 plus 2 plus 3 plus, and you go all the way to plus 9 plus 10. And I clearly could have even written this whole thing out, but you can imagine it becomes a lot harder if you wanted to find the sum of the first 100 numbers. So that would be 1 plus 2 plus 3 plus, and you would go all the way to 99 plus 100. So mathematicians said, well, let's find some notation, instead of having to do this dot dot dot thing-- which you will see sometimes done-- so that we can more cleanly express these types of sums. And that's where Sigma notation comes from. So this sum up here, right over here, this first one, it could be represented as Sigma. Use a capital Sigma, this Greek letter right over here. And what you do is you define an index. And you could start your index at some value. So let's say your index starts at 1. I'll just use i for index. So let's say that i starts at 1, and I'm going to go to 10. So i starts at 1, and it goes to 10. And I'm going to sum up the i's. So how does this translate into this right over here? Well, what you do is you start wherever the index is. If the index is at 1, set i equal to 1. Write the 1 down, and then you increment the index. And so i will then be equal to 2. i is 2. Put the 2 down. And you're summing each of these terms as you go. And you go all the way until i is equal to 10. So given what I just told you, I encourage you to pause this video and write the Sigma notation for this sum right over here. Assuming you've given a go at it, well, this would be the sum. The first term, well, it might be easy to just say we'll start at i equals 1 again. But now we're not going to stop until i equals 100, and we're going to sum up all of the i's. Let's do another example. Let's imagine the sum from i equals 0 to 50 of-- I don't know, let me say-- pi i squared. What would this sum look like? And once again, I encourage you to pause the video and write it out, expand out this sum. Well, let's just go step by step. When i equals 0, this will be pi times 0 squared. And that's clearly 0, but I'll write it out. pi times 0 squared. Then we increase our i. And, well, we make sure that we haven't hit this, that our i isn't already this top boundary right over here or this top value. So now we said i equals 1, pi times 1 squared-- so plus pi times 1 squared. Well, is 1 our top value right over here, where we stop? No. So we keep going. So then we go i equals 2, pi times 2 squared-- so plus pi times 2 squared. I think you see the pattern here. And we're just going to keep going all the way until, at some point-- we're going to keeping incrementing our i. i is going to be 49. So it's going to be pi times 49 squared. And then finally we increment i. i equal becomes 50, and so we're going to have plus pi times 50 squared. And then we say, OK, our i is finally equal to this top boundary, and now we can stop. And so you can see this notation, this Sigma notation for this sum was a much cleaner way, a much purer way, of representing this than having to write out the entire sum. But you'll see people switch back and forth between the two.