Properties
University of Missouri - Columbia
Telescoping


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hoping Siri's, which is a form of a geometric Siri's. And the best way to explain what this telescoping Siri's is is definitely by writing it out and showing it to you guys. So let's just look at an example. So let's say we have this song. So the Siris of one over, N minus one over and plus one and this Siri's is going from any holes, one to infinity. Okay, so we can kind of tell right off the bat. We have this edition. We have a subtraction of two terms. Geometrics not gonna be working here, but what we're going to do is start off is to just write out some of the partial sums. So the first partial some is just the first term in the Siri's, which would be one minus one half. So one minus one half would be the first term. The second partial term partial some term would be the first term of the Siri's, plus the second term of the Siri's. So when we plug into two, we get one half minus one third. The third partial some term would be the first term of the Siri's, plus the second term of the Siri's plus the third term of the Siri's, which would be when Ennis three, you'd have one third minus 1/4. Now let's say this keeps going. Let's look at what's happening here. If we were to drop the parentheses, this and this would cancel. We'd only be left with one minus one third. Here we would have one minus one half. Nothing would change here. One house would cancel and one thirds would cancel. And again, we'd only be left with one minus 1/4. That means when we get to our S of N when we get to Esa Ben, we're gonna have one. So every single partial, some has a one and that it also has minus one over and plus one. And it's too. When we have A s two, we have a three. When we have a three, we have a four. So now we have our s sub en, which is our partial son formula partial, some. So what telescoping is is when you write off the partial sums you on Lee end up with the very first term with the very first term. So the very first term was one minus one half, which was one. So we were on Lee left with one and Onley this piece of the formula of the original ace event and last. But it's not the last entire term. It's the last little teeny, tiny bit right here. So you only end up with the very first term in the last term. Again, if we were to write out the partial sums, we would have one half two thirds three force, which isn't going to give you a clear with pictures if you leave, it once implied. So let's say we wanted the seventh partial. Some the seventh partial song would be equal to a one plus a two plus a three plus a four plus a five plus a six plus a seven. That is what the seventh partial summons where these a sub ones are coming from up here and you would have to plug in a one, and then you would have to plug into two and then a three, and then you would have to add all those terms together but s up seven is also equal to one minus 1/7 plus one using the s sub end formula so one minus 1/8. This gives us a quick answer of one one minus 1/8 which is equal to 78 So if we were to add the first seven terms of the Siri's, we would get 7/8. But again, this is an example of telescoping because the middle terms are canceling as we go through our partial sums. Now we cannot figure out now. We can also figure out the limit as n goes to infinity of one minus one over and plus one. And the limit of this gives you one. And that means because because the limit as N is going to affinity of those partial sums is equal toe one, the Siris of one over n minus one over N plus one is equal toe one. So we have found that are telescoping Siri's converges, and we have found the some. The telescoping Siri's is very unique. You usually need these two terms being subtracted. That's the most common form you'll see it in. If you're unsure, you can always write out some partial sums and notice if some are canceling or not, and then you can come up with the partial. Some formula and take the limit of that to see what happens to your Siri's fraction minus of fraction. That's usually a pretty good indication of a possible telescoping Siri's. But to check, you need to start writing out the partial sums. The partial first partial. Some is the first term in the Siri's, which is when we plug in one for end, so we would get one minus one over route to the second partial. Some of the Siri's is the first term, thank you. Plus, when you plug in and is equal to two into the Siri's, which gives you one route to minus one over square root of three. The third partial song is the first two terms again. Notice that I'm not simplifying these. You will miss the telescoping Siris of you. Simplify. So third partial. Some is the first two terms plus, and then you have to add on that in equals three terms square root and well for Let's change this to a four, actually plucking in a three here. Um, I don't think we need to go any further. You can if you'd like to prove it to yourself more if we drop the parentheses, we would have one minus one over square to two plus one, divided by square to the minus, and the plus will cancel. Same with the third partial. Some thieves will cancel. Thes will cancel. So here I am left with one minus one over square to to hear. I'm left with one minus one over the square root of three. Then I am left with one minus one over the square root of four. So when I continue on and in my inthe partial some the pattern to get from 1 to 22 to the pattern between this and is equal to three and this one minus one over square root of four. I have a one minus one over and is three. Here I have a four that's N plus one, which matches with the last term in our Siri's. So now that I have what my end partial some is, I can take the limit as N is going to infinity of one minus one over the square root of end plus one. The limit of this right here is one that tells me that my Siri's converges and furthermore, it also tells me that I could write the some of the Siri's as being equal toe one. So this is another example of telescoping Siri's and how to use the end partial some to determine convergence or divergence being subtracted. Another indication of a possible telescoping Siri's. But we're going to start by writing the partial some so that we can check So the first partial some will be three minus 3/4 theseventies Partial. Some is the first term in the Siri's, plus the second term in the Siri's, which we confined by plugging in n equals. Two. When we plug in n equals two, we get three Force minus three nights when we go to find the third partial. Some term we have three minus 3/4 plus 3/4 minus three nights, and now we need to plug in and equals three, and we get three nights minus 3/16, and I don't believe we need to keep going. We can tell our 3/4 is canceling here. 3/4 cancels here. Then the three nights cancels, and we can go ahead and write our end partial term formula so you could tell if we right out to the side. We have three minus 3/4 and then we have here. We have three minus three nights. Here we have three minus 3/16. So if we need to find the ninth 10th, if we need to find the inthe partial some, the three is remaining the same. The three on top of the fraction is remaining the same. And here we have 12 becomes 93 is becoming the 16. If you're unsure, you can always look back to the Siri's. It's going to be this term in the Siri's that makes up the formula for the partial. Some. So we have three divided by and plus one squared. Always a good idea to check. Let's see if we can find the fifth partial. Some so s five would be equal to three minus three. Over. Well, what's not? Yeah, this will work 33 minus 3/4 squared. This is equal to three minus 3/16. And if you were to write out s four and s f five the way we're doing this is what you should end up with. So now that we have found our inthe partial, some and our end partial. Some is equal to three minus three over and plus one squared. We can take the limit as N goes to infinity of our partial some. And when we do this, this term right here is going to zero because the fractions getting very, very small. So our limit is equal to three now that we know are limit of our partial. Some we could say that our Siri's, which was three, divided by n squared, minus three, divided by N Plus one squared, is equal to three and that it converges. So not only do we know that some of the Siri's we also are able to say that the Siri's convergence and it does converge to three again. This was looking at a telescoping Siri's we can use, um, telescoping Siri's. It also doesn't look like a geometric Siri's, because we do have this plus one, however, this can. This right here can be rewritten as one over in minus one over and plus one. Now, this Siri's. If we put the signal notation here and this is telescoping and we have already found this, but we'll do it again a little bit quicker. We write out the first partial term by plugging in and is equal toe one. We get one minus one half. I'm gonna do it without the parentheses. Thesis. Econ Partial. Some is one minus one half plus one half minus one third and the third partial. Some is one minus one half plus one half minus one third plus one third minus 1/4 and you can see what's happening here. And if we write out our sequence of partial sums, we get one minus one half one minus one third, one minus 1/4. So our and partial some will be one minus one over and plus one again. If you're not sure where this is coming from or if you can't tell you can always go back to your Siri's, it's going to be the last part in the Siri's. And if we take the limit as N goes to infinity of our partial sums, we get a limit of one, and that tells us that our Siri's here's our original. Siri's is equal to one, and it converges so you can rewrite this. Siri's as a telescoping Siri's. We did this using partial fractions. We're gonna use techniques coming up, though, that we won't have to do something like this. It will be something that comes much more natural. I just wanted to point out an example where you didn't have this subtraction immediately. But a telescoping Siri's is almost always given away by a fraction minus another fraction. And then you would find the partial sums, find the formula for the partial sums for the sequence and then take the limit to figure out if you're serious, converges or diverges.