Siri's just so we can handle some Siri's that might look a little bit different than what we've been seeing so far. So some properties of the Siri's Let's Say we have a Siri's a seven with a some of A and another Siri's B seven with a sum of beat. So if we have this, this is given. So if this then we can say a couple of things about some, um, properties of this. So the first thing we can say is that if we have a seven plus B seven, so if we have a Siri's with two functions added together like this, this summation sign, we'll go through the addition and we can take the sum of a sub in, and we can take the sum of B seven and add them together so this would be a equal to a plus B. You can treat thes Siri's individually. You can also do the same thing if you have a Siri's with two things being subtracted, this will be equal to the sum evasive in, and you can subtract the some of the seven, so this will give you a minus. Bi another way to handle telescoping. Siri's right here so you don't have to use partial sums. You can evoke this property number two for number three. The last thing we can look at is if we have a summation and we have some number regular, constant number K times the syriza a sub in. We can pull the K out of the summation, and we're just left with the summation of a seven. This would be equal to K times a So these air some properties again. Number two. This can be used if you recognize it as a telescoping Siri's instead of finding partial sums. So we'll do an example where we use this instead of partial sums. This can be used occasionally. Okay, so now we'll look at some examples of using these three properties when we're looking at. Siri's also have to serums that we want to talk about when it comes to Siri's. And we're gonna introduce these two, the're, um, So the first, the're, um, or big property. On top of what we just talked about is that every non zero non zero multiple of a Siri's every Nazeer multiple of a divergent Siri's diverges. So what? This means is that if you have a Siri's that diverges and then you multiply by five, this Siri's will still diverge. So every non zero multiple of a divergent Siri's diverges the second tier, Um, or thing we're going to look at is that if a Siri's a sub in converges, so if a suburban converges and we have a Siri's beasts of been that diverges So we have one Siri's that converges one, Siri's diverges. Then, if we take the sum of the two added together, then this or if we do the same thing with subtraction diverge. So if you have a convergent plus or minus a divergent Siri's, the some or those two. If we combine them, we're going to diverge. So we're also going to see some examples of these two therms in action when we're trying to say converge or diverge. Answer that question about a Siri's plus one divided by three day land. So forgiven this we can go ahead and break this up. So this is going to be equal to the Siris of five over to to the end, plus the Siris of one divided by three to the end. So we're seeing that addition property, which was Property one in action. Now we can look at each of these individually to decide something, so we have the Siri's 105 over to to the end. If we write out a couple of these terms for this Siri's, we get five, and then the next term is five halves. The next term in the Siri's is five force. The next term in the Siri's is 58 To get from 5 to 5 halves, you multiply by one half to get from five halves to five. Fourth you multiply by one half and 5/4 258 you multiply by one half. This tells me that I have a geometric Siri's with a equal to five and are equal toe one half. That means my Siri's here converges. Now if I look at the other half of this Siri's, I have the Siris of one divided by through to the end, from an equal zero to infinity. This is equal to If you plug into zero, you get one. If you plug in a one, you get one third and then we have one night one 27th, and this would continue. So here we have a geometric Siri's with a equal toe. One in our our is one third. So this Siri's also converges. We have a convergence, Siri's plus a convergence Siri's. So this, Siri's added together should converge. But we can also find the some of this entire Siri's by adding together the some of these two individually. So to find the some we have five. A over one minus are, which is one half and five divided by one minus. One half is equal to 10. And then over here we have one minus one minus one third, and this gives me three halfs. So now if I look at this, I have this Siri's. This was my original Siri's five divided by two to the end, plus 1/3 to the end. I broke this up into the sum of five over to to the end, plus the sum of 1/3 to the end. I said this Siri's right here was geometric, and it's some was 10. This Siri's right here was geometric, and the sum was three halves. And if I add these together, this gives me 23 over, to which is also the some of the our original entire Siri's, and this does converge. So what we've just shown is that if you have something like this as your original, you want to go ahead and split up the Siri's by moving by using the first property that we could take the Siri's of the first function, plus the Siris of that second function, and then add those together to find the some of the original entire Siri's, how much we can do with it. We're going to do some algebra to try and simplify it first. So instead of looking at, this is one big fraction. We're going to use our trick and split the fraction. So we're gonna have three to the end, minus one divided by six to the end, minus one minus 1/6 to the end, minus one. This is equal to the Siris of one divided by two to the end, minus one minus 1/6 to the end, minus one. So now that we have this, we're gonna use property to that we talked about. And we're gonna make this the Siris of one over to to the n minus one minus the Siris of 1/6 to the end, minus one. So now what we can do is we confined the some of this Siri's that some of this Siri's do the subtraction and will have the some of the original Siris. Both of these Siris are in fact, geometric. That index on all of these is N is equal to one. And if you're not convinced their geometric, we can write out a few terms. So if N is equal toe one, we get one divided by two to the one minus one, which is zero. So we get one plus plugging in a two, we get one half. If we plug in a three, we get 1/4 and this will keep going. This tells me my a is equal toe one, my r is equal toe. One half the some of this will be a over one. Minus are which is equal to 1/1 minus one half, which is to so so far I have to which is the sum of the first Siri's. Now I need to do the second. Siri's right here. Same thing If we plug it in one. I get one plucking in a two. I get 16 clocking in a parking in a one, I get 16 plugging into two. I get 16 plugging into three. I get 1/36. This would continue. My A is equal toe one, my r is equal to 16 I'm always going to be multiplying by 16 That means my some is 1/1 minus 16 which should give me six. Fifth, 6/5 at 6 50. Aiken, go up here and subtract. So I have the first part, minus the second part. So SEC first, some minus second some. This gives me 4/5 and that's my final answer. So what this means is that the summation of the original Siri's that we were looking at before we split it? We can say that this Siri's some is for fifth, and we can also say that it converges. So this is what you want to be able to say. We were able to say this because we split up the integral split up the summation, the Siri's. And then we found each Siri's individually as a geometric. Siri's the end. So what? We're going tohave is we can use the third property, pull the four out because it's a constant and we have the Siris of one divided by two to the end. Now we need to find this. Siri's right here definitely looks like it could be geometric, but we'll write out a few terms to see we plug in zero, we get one. We'll have one half been 1/4 then 18 and this will continue. We have an a equal toe one, and in our equal toe, one half every single time. You're multiplying by one half to get the next term. To find the what this some equals we use 1/1 minus are, which gives us two. Now we can say that four times to which is eight is the some of the original Siri's. So we have that the summation of four divided by two to the end from an equal 02 Infinity is equal to eight and that this Siri's converges minus and divided by end to to the end. I've also given you this note, which is that the Siri's one over and diverges we will eventually prove this, but from for where we're out right now. I'm giving you this information. So let's get back to our Siri's. The first thing we're gonna do is use our trick and split the fraction. So we're gonna have to to the end over end to to the end, minus and divided by end to to the end with some algebra and some canceling Here and here we have this summation of one over end minus summation one over to to the end and I've gone ahead and brought my Sigma through the minus sign. I've already told you that this right here diverges So this Siri's diverges this Siri's the one over to to the end. We actually just found that this is a geometric Siri's with rto equal to one half, which means the Siri's converges. So what we have now is a Siri's that diverges minus a Siri's that converges. But by one of the second, the're, um that we talked about because we have a Siri's that diverges minuses. Siri's that converges. The summation of both of them put together is going toe also diverge. So that means the final answer for this this Siri's would diverge because we have a diverge minus a converging Siri's when we split it up. So this right here this would diverge. Yeah, using the properties, we can split this into two Siri's one over in. I've already told you that this diverges and will prove this at a later time, but this diverges. So now we have two times the Siri's that diverges, and the property that we talked about said any multiple of a divergent Siri's is also going to diverge, and this is an example of that. So this uses the serum that tells us a divergent Siri's multiplied by something will also diverge.