Let’s learn a little bit about sequences and series. So what’s a sequence? Well, a sequence is just a bunch of numbers in some order.
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Sequences and Series (part 1)
You know, the most difficult sequence is 1, 2, 3, 4. You get the point. And what’s a series? Well, it’s often represented– it’s just a sum of sequences, a sum of a sequence.
So, for example, the arithmetic sequence– sorry the arithmetic series is just the sum of the arithmetic sequence. So 1 plus 2 plus 3 plus– we could keep going until maybe some number. This is called the arithmetic series.
Nothing too fancy here. But before we move forward, let’s get a notation for how we can represent these sums without necessarily having to write out all of the digits or having to keep doing this dot, dot, dot, plus notation. And that notation is Sigma notation.
That’s an upper case Sigma. And how do you use Sigma notation? Well, let’s say I wanted to represent this arithmetic series. So I would say, well, let’s add up a bunch of– let’s call them k’s.
This is an arbitrary variable. And we’ll start at k equals 1. We’ll start at k equals 1, and we’ll go to k is equal to big N. And this is the exact same thing, so we first make k equal to 1, and then we add it to k is equal to 2 plus 3, and we go all the way until N minus 1 and then plus N.
So this is the Sigma notation for the arithmetic series. Before I move on, I think this is a good time just to like learn a little bit more about the arithmetic series. We’ll actually focus on this one and the geometric series because those are the two that you’ll see most often.
And then once you learn calculus, I’ll show you the power and Taylor series, which is exact– the Taylor series is a specific version of a power series. But let’s play around with this arith– I keep wanting to say arith-MET-ic, but a-RITH-metic, either way– series. So let’s call the sum S. Let’s say that this is equal to the sum from k is equal to 1 to N of k, which is equal to, just like we said, 1 plus 2 plus 3.
And we’ll just keep adding them, dot, dot, dot, to a bunch of numbers, to big N minus 1 plus big N, right? Fair enough. Now, bear with me a second. I’m just going to write that same exact sum again, but I’m just going to write it in reverse order. And I think it’s intuitive to you that it doesn’t matter what order I add up numbers in.
They’ll add up to the same number. 2 plus 1 is the same thing as 1 plus 2, right? So let me write this exact same sum, but I’ll write it in reverse order. So that’s the same thing as N plus N minus 1, plus N minus 2, plus– and the pluses keep going– plus 2 plus 1. This is the exact sum, just in the reverse order.
And I did that for a reason because now I’m going to add both sides of this equation. I’m going to take– S plus S. Well, that’s just 2S. And that’s going to equal this sum plus this sum.
I wrote this so that the sum becomes clean. And why do I say that? Well, let’s add up corresponding terms. We could have added up any terms, but– so since they all have to add up, let’s just add the 1 plus the N, then we’ll add the 2 plus the N minus 1, then we’ll add the 3 plus the N minus 2, and so forth.
And I think you’ll see in a second, or maybe you already realize why I’m doing this. One plus the N, the 2 plus the N minus 1, the 3 and the N minus 2, all the way to the N minus 1 and the 2, the N and the 1. What’s 1 plus N? Well, that’s just N plus 1, right? What’s 2 plus N minus 1? Well, that’s also N plus 1, right? What’s 3 plus N minus 2? I think you could guess.
It’s N minus 1. And we just keep doing that. And what’s N minus 2 plus 2? Sorry, this is a plus. N plus 1.
And what’s N plus 1? Well, that’s just N plus 1, of course. So my question to you is how many of these N plus 1’s are there? Well, there are N of them, right? Each N plus 1 corresponds to each of these terms, so there are N of these. So instead of just adding N plus 1 N times, we could say that this is just N times N plus 1. So we have 2 times the sum is equal to N times N plus 1, and we could divide both sides by 2, and we get the sum is equal to N times N plus 1 over 2.
Now, why is this neat, or why is this cool at all? Well, first of all, we found out a way to sum this Sigma notation up. We got kind of a well-defined formula. And what makes this especially cool is you can use this for low-end parlor tricks.
What do I mean by that? Well, you can go up to someone and you can say, well, how quickly do you think I can add up the numbers between 1 and– what am I doing– oh, between 1 and 100? And, you know, people will say, oh, it will take you a little time: 1 plus 2 plus 3. And you say, well, it takes me no time at all because this is what I can do. So the sum– and I just want to show you that you can use different variables from B equals 1, we’re taking the variable B, to 100, right? That’s the sum from 1 to 100.
And we figured out what that formula is. It’s going to be 100 times 101 over 2. Well, what’s 100 times 101? It’s just going to be 101 with two zeroes, right? 10,100 over 2, and that equals 5,050.
That’s pretty neat. Instead of having to say 1 plus 2 plus 3 plus blah, blah, blah, blah, blah, blah, blah, blah, plus 98 plus 99 plus 100, this would take you some time, and there’s a very good chance you would make a careless mistake. We could just plug into this formula, which we proved and hopefully you understood, and say that equals 5,050. You could do even something more impressive: the sum from 1 to 1,000.
What’s the sum from 1 to 1,000? Well, our formula, remember, was N times N plus 1 over 2. So if N is equal to 1,000, then what’s our sum? It’s 1,000 times 1,001 over 2, which is equal to– well, we’ll just add three zeroes to this: 1,001, one, two, three. Sorry, I think that was my first burp ever on one of these videos.
I should re-record it, but I’m going to move forward. That kind of disconcerted me a little bit. I’d eaten too much.
Anyway, divided by 2, and what is that? Let’s see, it’ll be 500– let’s see, this is a million. Half of a million is 500,000. 500,500.
And that would have taken you forever to do manually. But based on this formula we just got, you know how to do it very, very quickly. So that’s the arithmetic series.
But let’s do another one. This is another typical series that you might see. Actually, this one you’ll see a lot in your life, especially if you go into finance or really a whole series of scientific– this shows up a lot, and this is called the geometric series.
And the geometric series is– essentially you take x. And I’ll do it generally where I just take a variable x, and I say– well, no, no. Let me just not take an x. Let me just take some number.
So let’s say some number a to the k from– I don’t know. Let’s say from k is equal to 0 to k is equal to N. What does that mean? Well, that means a to the 0, right, k is 0, plus a 1 plus a squared plus a to the third plus– and you could keep going– plus a to the N minus 1 plus a to the N minus 2.
This is called the geometric series. And it might not be obvious to you, but this type of growth, where you keep increasing the exponent, this is called geometric growth. So how do you take the sum of this? Well, let’s see if we can do a similar trick, although this trick will involve one more step. Let’s call it the sum from k equals 0 to N, a to the k.
And that, of course, is equal to what I just wrote. I probably didn’t have to do it like this. a squared plus bup, bup, bup, bup, plus a to the N minus 1, plus a to the N minus 2. Now let’s define another sum, and I’m going to call that aS.
Actually, I’m about to run out of time, so I’ll continue this in the next video.
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