Defining convergent and divergent infinite series
Partial sums intro
- [Voiceover] Let's say that you have an infinite series, S, which is equal to the sum from n equals one, let me write that a little bit neater. n equals one to infinity of a sub n. This is all a little bit of review. We would say, well this is the same thing as a sub one, plus a sub two, plus a sub three, and we would just keep going on and on and on forever. Now what I want to introduce to you is the idea of a partial sum. This right over here is an infinite series. But we can define a partial sum, so if we say S sub six, this notation says, okay, if S is an infinite series, S sub six is the partial sum of the first six terms. So in this case, this is going to be we're not going to just keep going on forever, this is going to be a sub one, plus a sub two, plus a sub three, plus a sub four, plus a sub five, plus a sub six. And I can make this a little bit more tangible if you like. So let's say that S, the infinite series S, is equal to the sum from n equals one, to infinity of one over n squared. In this case it would be one over one squared, plus one over two squared, plus one over three squared, and we would just keep going on and on and on forever. But what would S sub -- I should do that in that same color. What would S -- I said I would change color, and I didn't. What would S sub three be equal to? The partial sum of the first three terms, and I encourage you to pause the video and try to work through it on your own. Well, it's just going to be the first term one, plus the second term, 1/4, plus the third term, 1/9, is going to be the sum of the first three terms, and we can figure that out, that's to see if you have a common denominator here, it's going to be 36. It's going to be 36/36, plus 9/36, plus 4/36, so this is going to be 49/36. 49/36. So the whole point of this video, is just to appreciate this idea of a partial sum. And what we'll see is, that you can actually express what a partial sum might be algebraically. So for example, for example, let's give ourselves a little bit more real estate here. Let's say, let's go back to just saying we have an infinite series, S, that is equal to the sum from n equals one to infinity of a sub n. And let's say we know the partial sum, S sub n, so the sum of the first n terms of this is equal to n squared minus three over n to the third plus four. So just as a bit of a reminder of what this is saying. S sub n... S sub n is the same thing as a sub one, plus a sub two, plus you keep going all the way to a sub n, and that's going to be equal to this business, n squared minus three over n to the third plus four. Now, given that, if someone were to walk up to you on the street and say, okay now that you know the notation for a partial sum, I have a little question to ask of you. If S is the infinite series, and I'm writing it in very general terms right over here, so S is the infinite series from n equals one to infinity of a sub n, and the partial sum, S sub n, is defined this way, so someone, they tell you these two things, and then they say find what the sum from n equals one to six of a sub n is, and I encourage you to pause the video and try to figure it out. Well, this is just going to be a sub one, plus a sub two, plus a sub three, plus a sub four, and when I say sub that just means subscript, plus a sub five, plus a sub six, well that's just the same thing as the partial sum, this is just the same thing as the partial sum of the first six terms for our infinite series. It's just going to be the partial sum S sub six. And we know how to algebraically evaluate what S sub six is. We can apply this formula that we were given. S sub six is equal to, well, everywhere we see an n, we replaced with a six, it's going to be six squared minus three over six to the third plus four, so what is this going to be? Six squared is 36 minus three, so that's 33, and six to the third, let's see, 36 times six, I always forget, my brain wants to say 216, but let me make sure that that's actually the case. Six times 30 is 180, plus 36, yes, it is 216, so I guess I have, inadvertently, by seeing six to the third so many times in my life, I have inadvertently memorized six to the third power, never a horrible thing to have that in your brain. So this is going to be 216 plus four, so 220. So, S sub six, or the sum of the first six terms of the series right over here, is 33/220, and we're done. And the whole point of this is just so you kind of appreciate, or really do appreciate this partial sum notation, and understand what it really means.
AP® is a registered trademark of the College Board, which has not reviewed this resource.