Geometric Series
Telescoping
Properties
University of Missouri - Columbia
Partial Sums


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Video Transcript

please now. So what we just talked about in the previous lecture, video and topic was sequences where sequences were a list of numbers. So a Siri's is in the form a one plus a two plus a three plus a four plus dot dot dot. So a sequence had the commas but a Siri's. We're adding these numbers now. It is a some. So this form right here is what we're going to call an infinite Siri's. It goes on forever. So here we have an infinite Siri's Infinite definitely has an E on the end, and this is also can be equal to the Sigma and equals one to infinity of some a seven. We're a seven. That is the formula to get the Siri's just like we had the formulas to get the sequences. The only difference is now we're adding the numbers together instead of just listing them. So formula. Yeah, for the Siri's thank you. So this is the notation that will be using for the Siri's um, a sub en is the 10th term. So this is the 10th term. A one would be the first term h you would be the second term So now that we have, this is the set up for a Siri's. So if we look at a Siri's a one plus a two plus a three plus a four and its infinite, it continues to go on. We have something called partial sums of a Siri's, so the partial sums for a Siri's look like this the first partial. Some is just the first term, the second partial. Some is the first term, plus the second term, the third partial. Some is the first term, the second term, then the third term. And this continues. So for partial sums, you're just adding a certain number of terms from your Siri's. The 10th Partial. Some is a one plus a two plus. This should be your entire up to a seven Siri's and this is K equal one to end of a sub K. So this is how we do know the sequence of partial sums of the Siri's s A Ben is the partial Some and you would list these S s as a sequence. So you would list this as s one as to as three all the way toe s n. And this is what we're going to define as a sequence off partial sums sequence of partial sums. Okay, so now we need to talk about the relationship between sums and partial sums. So we have our Siri's, and we say that the some of the Siri's we are denoting that n equals one to infinity of ace events. So this would be our Siri's. Every Siri's has a partial some. Here's the relationship, if the sequence. Because remember, we listed out the partial sums with commas that makes it a sequence. So if the sequence of partial sums has a limit, L, um, has a limit l and this partial. Some sequence converges to L then. So once we have the sequence of partial sums and we found the limit, then we can say that the Siri's. So now we're saying something about the whole Siri's, then the whole Siri's. Not only does this whole survey Siri's converge to l converge to l, but the sum is l. So then the Siri's converges to L. And the sum is l. And the some of the Siri's is l. So what this looks like is now we write a one a two. So here's our Siri's written now and this keeps going taste event and then would keep going. And we already said that this is also equal to a seven as an equals one to infinity. But now, if we know the limit of the partial sums, we can say that this infinite Siri's we can give it a finite number as the some and we call that ill. So in summary, what we can say is that the sum of a Siri's is the limit of the partial sums is the limit of the partial sums. And this is very important because one of the main questions that we're gonna be looking at is how do we find an infinite sum? This is one way to look at the partial sums the other relationship that we have that we should mention. We've mentioned what happens if the partial sums converge. But if partial sums diverge. So if the partial sums diverge, yeah, then we can also say the Siri's diverges. So again, if we go back to the intro video, the main question that we're asking is, how do we know when a Siri's diverges or convergence? So we've answered this one way we can use partial sums, and we've discussed the partial sums and how they work and what the relationship is. So now we're going to see an examples of how to use the partial sums and a little bit more about exactly what they are. We're going to look at the Siri's that is as n goes from one to infinity of two, divided by three to the end, minus one. This will be equal to. So let's expand this sigma notation plugging in a one. We get to over three to the zeroth power, which is to over one which is to again we need to use plus signs were adding, so we would have to plus and then we would plug into two on. We would get two thirds because we have three squared on the bottom. Then we would have We would have three to the first power. Then we would have three squared on the bottom, which would give us to ninth. Then we would have three cubed on the bottom for 2/27 this would continue on. And now we wanna look at if the Siri's converges or diverges and we wanna look at the partial sums, So the first partial some is to the second partial. Some is two plus two thirds, which is equal to 8/3. The third partial. Some is the first three terms two plus two thirds plus two nights. This is equal to 26/9, the fourth partial. Some is equal to two plus two thirds plus 2/9 plus 2/27 this is equal to 80/7. This would continue and you would eventually see that you're inthe partial. Some is equal to And then you would write a formula. So the end partial some. I'm gonna write it up here so we can see it better. The end partial. Some of this particular Siri's is two times one minus one third, the one third to the inthe power, all of it divided by one minus one third. So let's look at this. If you plug in a one to this function, you get one minus one third, which is two thirds. You get four thirds divided by two thirds, which would give you to because the threes would cancel for divided by two. So this is the secret. This is this equation for this sequence of partial sums, which would be written as to eight thirds 26/9. 80/7. So here, up top, we have our Siri's. So this is what we were given and all of this down here below is what we're going to call the partial some. Okay, now partial sums will keep going. And now we want to answer. The main question is, does the Siri's converge or diverge? So we know we have a partial some formula that's equal to two times one minus one third to the end, power divided by one minus one third. And we want to know something about the Siri's so we can take the limit as n goes to infinity of s seven, which is equal to the limit as n goes to infinity of two times one minus one third to the end, Power divided by one minus one third. Okay, so now that we have this, we can take the limit. So the limit here, this is going to be equal to choose a constant going to stay the same one minus. This is 1/3 and we can bring the end inside, divided by two thirds. So here this bottom part of the fraction is getting larger and larger and larger. But what this means is that this fraction is going to zero because the denominators getting infinitely large. So now what we have is two divided by two thirds or just two thirds. So what we have is two divided by two thirds, which simplifies to three. So now what we have is that the limit as n is going to infinity of S m N is equal to three. So this is the limit of our partial some. So this is the partial, some in the limit. So now what we can say is that the Siri's as n goes from one to infinity of our Siri's, which was to over three to the end, minus one is equal to three, and it converges. So now not only did we answer our Siri's converge, we also found the some. So the limit of the partial some is the actual Siri's total. It's what the Siri's sums to so that some of our Siri's is three, and we use the limit of the partial sums to find that partial sums. It's a series of one divided by N plus one times in plus two. If we write out a couple of terms in our Siri's, we have one divided by two times three. Because the equation for the Siri's isn't given to me simplified, I'm going to write it out on Simplified. Then I have one divided by when n is, too. I have three times four plus one divided by four times five plus one, divided by five times six, and this would keep going on. So now we can look at our partial sums, and instead of looking at the partial sums here, I'm just gonna go ahead and give you what the formula is for the partial sums, The partial. Some formula is one half, minus one over and plus two. But we're asked, Is this Siri's converge? Or does this Siri's diverge? And to answer this question using partial sums, we're just going to look at what happens right here. Okay, so this the limit as an goes to infinity of our partial some, which is one half minus one over and plus two. One half is a constant, So this is one half times the limit as N is going to infinity of negative one over and plus two here we can see that the bottom of this fraction is going to be growing very, very, very, very large. And this will in fact go to zero. So we have one half minus zero, which means our limit is one half. This is equal to l. So now we have said that the limit as N is going to infinity of our partial. Some function which was right here, is equal to one half. So now we can say that our Siri's of one over and plus one times and plus two the sum of our Siri's is also equal toe one half and we can say because we have a finite some here that are Siri's converges. So again, here's another example of using the limit of a partial some to make a decision about the entire original Siri's