Telescoping
Properties
University of Missouri - Columbia
Geometric Series


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which is called a geometric Siri's. And this is a very well known Siri's, and it is common. So a geometric Siri's so geometric Siri's is in the form a plus, a R plus a R squared plus a R cube plus, dot, dot, dot and in summation form. This is a to the r to the N minus one where n is equal toe one and then we go to infinity on this and it can also be any hole zero. This would just be a to the art, and you wouldn't have that minus one there. But that's okay. The point of the geometric Siri's is your starting here. So this is your start. You start at some number. In our case, it's a and then you multiply by our to get your next term. But then, to get your next term, it's also times are again. And then from here to here is times are again. So you're always multiplying by the same thing, multiplying Bye, the same thing. And this is what it means to be a geometric Siri's. So if you're always being multiplied by the same thing from number to number, that is when we're going to use a geometric Siri's So this are a is the first term in the Siri's A is your first term, and we call our the ratio. The ratio can be equal to one. It could be less than when it could be positive. It could be negative. Um, it doesn't matter. We call this the ratio, and it can be positive or negative. So this is the set up for a geometric Siri's. This is what it should look like. You're always multiplying something by the same thing. Now here is the part again are the main question that we're trying to answer is, Does a Siri's converge? Or does the Siri's diverge and a geometric? Siri's helps us answer that because N a geometric Siri's. So let's say we already have a geometric Siri's. We've identified it as a geometric Siri's. So this is in the form. A times are to the end. If end is going from zero to infinity now, okay, so in a geometric Siri's where we have this are so if the absolute value of our which is our ratios, what we're multiplying by, if it is less than one, the Siri's converges, and the answer is one of our main questions Now on Lee. Can we answer if the Siri's converges? But we can also say we can also say something about the some of the Siri's and the some, as then goes from one to infinity of a are to the n minus one. So the some of the Siri's is equal to the formula. A over one minus are so we can also say something about the some. So the sum is a over one. Minus are So now on Lee, Can we say that the Siri's converges? But we can also say something about the some. This is a unique feature of the geometric Siri's again we're looking at are in the geometric Siri's. The other thing we can say is that if that absolute value of R is greater than one, if it is greater than one, we can immediately say that our entire Siri's diverges. So now we can see why this is so helpful. The Siri's diverges and just to recap and so you guys can see it on the same page. If R is greater than one, not only can we say the Siri's converges, but we can say that the Siri's can verges to a over one. Minus. R R is the ratio that's being multiplied. This is if our is less than one. This series converges are is your ratio is always your first term, and now we're gonna look at some examples where we're going to see a Siri's, identify it as geometric, find a find our and then we can make a decision on whether the Siri's converges or diverges give you the information that a is 1/9 R is equal to one third and that we do have a geometric Siri's. So we're gonna just really start this one from the beginning, so a first term always are is always your ratio. So when we go to write out the Siri's, we know that our first term is 1/9 and our second term is 1/9 times are ratio. Plus, our second term is this 1/9 1 3rd. So we have 1/9 and then we have one third again. But then we multiply by one third again. So this is what it should look like. So you have one night plus 1/27 plus 1/81 plus and your next term would be won over. So from one night toe one 27th, we're multiplying by one third, 1/27 1/81 were multiplying by one third. So to get the next term, it would be one over and then 81 times three, which is to 43 plus, and then this would keep going on. Okay, this is equal to thesis Imation of 1/9 times, one third to the N minus one if we're starting at n equals one and going to infinity. So this is what our geometric Siri's would look like. Now what we what can we say about the geometric? Siri's? Well, R is equal toe one third, which is less than one. So we know that our Siri's should converge and we also know that are seriously should converge to a divided by one. Minus are, which is equal to 1/9 divided by two thirds, which simplifies toe 1/6. So not only do we know our Siri's converges, we also know it converges to 16 This means this Siri's we can write it as being equal toe 16 This is the sum of the Siri's. This is what the Siri's converges to. Let's look at another example. Now start by writing out a couple of terms. If we plug in and is equal to zero, we start with a five. Plugging in n equals one gives you negative 5/4. Plugging in n equals two gives you 5/16 and equals three. Gives you 5/64 this is alternating. So this should be a minus right here, and this pattern is going to continue. Now that we've written on the first few terms, Let's look and see if we can figure out a pattern. So to get from 5 to 5 force, you have to multiply by 1/4. But you also have to change it to a negative. So five times negative 1/4 gets you five over negative 5/4 to get from negative 5/4 2 plus 5/16. We need a negative because negative and a negative will change that sign. 5/4 Thio 5/16 5 times five You need a one 4 to 16. You need a four, so we're looking good. So far. Let's check one more term To get from a plus to a minus, you're gonna need a minus. And then again, it's gonna be 1/4. So every time you're multiplying by negative 1/4 this tells us that we have a geometric Siri's with our first term equal to five and our our is equal to negative 1/4. Because that's what we're multiplying by every single time immediately. Because the absolute value of R is equal to the absolute value of 1/4 which is less than one, this automatically means that are serious converges. So we're able to tell that right off the bat. Furthermore, we can use that a over one minus are, which is equal to five divided by five force, which will give us four. And this is the sum of the Siri's some of Siri's so again, not on Lee. Can we say that the Siri's converges. We can also find the some of the Siri's by using this geometric. So this means that the some of negative one to the end times five, divided by four to the end, is equal to four. So the Siri's converges and the some of the Siri's is four, and this is all based on the fact that we had a geometric Siri's with a ratio less than one about it back four to the end. Start by writing off a few terms At N is equal to zero, our first term in the syriza's one. Then we have a negative 1/4 in a positive 1/16 and this is going to keep going and pretty immediately, based on the form of this and the couple of terms that we've seen to get from a one to a negative 1/4 you multiply by negative 1/4 to get from a negative 1/4 to a positive 1 16. You again multiplied by negative 1/4 and we're pretty much can go ahead. And we're safe to assume that we do have a geometric Siris are a is one, and our our is negative 1/4 again right off the bat. The absolute value of our is less than one, so our Siri's does in fact converge okay, and furthermore, if we use that one divided by one minus are, which is equal to one divided by five force, which is equal to 4/5 we can say that the Siri's negative one to the end, divided by four to the end, is equal to 4/5 so we can also find the some of the Siri's two Be 4/5. Again, this formula is a divided by one. Minus are. This is the formula that we're using toe. Identify something as geometric quickly. I like to look for the exponents. So if both are raised to the end, if it appears that it's gonna be a problem if either that if there's no addition, if there's no addition, if it's on Lee multiplication or something raised to the power, that's another really good clue that it's gonna be geometric. So look for no addition and you're me, right? This a little bit better. Look for that. No addition. These air not tell tale absolutes, though, but you wanna look for having no addition, and you wanna look for exponents. That's usually a good sign as well. And again, really, the best way to tell is to write out the terms and see if you can quickly identify what's being multiplied to get from one term to the next. But it does have to be the same thing multiplied every time, no addition. And we do have exponents, so that's a good sign that it's gonna be a geometric Syrians. But I would still encourage you to write out the first few terms. So if we go to write out the terms we plug in and is equal to zero, we get to and and is, equal toe one will give us four. Fifth and equal to two will give us two cubed on top, which is eight divided by five squared, which is 25. Plus, Let's write out one more we would get to Cube, which is two times two is +44 times to his A eight times to a 16. Too cute divided by 1 25. Okay, and this would continue. So this is the first few two terms of our Siri's. So let's see, how do we get from 2 to 4/5? You would multiply by 2/5 to get from 4/5 2 8/5 you and multiply four by two. Four times two is eight and you and multiply five by five to get from eight fists to 16 to get from 8 to 16 you would again multiplied by two, and then you would multiply that 25 by five so you could see that we're multiplying every single time by 2/5. This is the set up for a geometric Siri's, with a being equal to two in our are being equal to 2/5 and because are is less than one. This Siri's converges. So we have a geometric Siri's that converges because our is less than one. And not only can we say that the Siri's converges, we can also say that the some of the Siri's to to the end plus one divided by five to the end, is equal to a divided by one minus are, which is equal to two divided by one minus 2/5. And when you you can use your calculators and mental map, this gives you 13th as the sum. So this right here is the sum of the Siri's, and we also know that the Siri's converges. So we have the some of the Siri's again, another geometric Siri's. It is very important to ride out the first few terms to get an idea of what's going on before you immediately assume it's geometric real quick. Ask you the question. Do these two Siri's that I've written? Do they converge or do they diverge? And these should be quickly. If you've been paying attention, converge or diverge right off the bat for Let's call this one a one times 2, 50 times to 50 times to fit. So every term is being multiplied by 2/5. That makes our our 2/5 which is less than one. Immediately. We can say that the Siri's converges. There's no other work that we need to do here. Down here for B. It's a little bit harder for me to see what's being multiplied. So I'm going to go ahead and simplified. I have one minus three plus nine, minus 27 plus dot, dot dot one to negative three. You would multiply by negative three. To get from negative 3 to 9, you would multiply by negative three. To get from nine to negative 27 you would multiply by negative three. So now I can clearly see R R is equal to negative three. The absolute value of negative three is greater than one, and we can immediately say that our Siri's diverges So again, if you're just asked the question, converge or diverge, you just need to look at our if you're asked for what the some of the Siri's is, if you're asked for so we could be asked, converge or diverge. We could also be asked the some. If you're asked for the some, that's when you need to use the A Over one minus are, but you can Onley fined the sum. If our is less than one, that's the only time this is going to make sense.