So we have this nice looking function. I think this function is a very friendly function. Friendly function. Friendly friendly in quotes. Let me just take it off. So X squared over one minus X cube. Okay, so we want to work on this thing. We want to find a power serious representation of just this guy. Once we do that, we're just gonna multiply by this Moon reader. And then we're done. Also we already know this format. We really know this one. So you can see that this one looks like this one is just that. It has a cute. So in order to get this cute, I think you have to differentiate this one twice. Okay. And definitely the first time you're gonna have squared and differently. The second time you're gonna have twice. Okay, so whenever I differentiate this one twice and I'm just gonna have this one. Now, all that I have to do is differentiate this one twice. What is the derivative twice derivative of this one? It's just gonna be in from 2 to infinity. Uh and in -1. Hoekstra Power in -2. That is the second derivative. And that's it. God is the power of serious representation. Okay. Uh So we can go on and find the radius of conversions. Oh I beg your pardon, That is not a part of. So we supposed to multiply by this guy. So whenever we multiply by X squared plus X, then it's just most find this one by X plus X squared. So what we have is in the face equals uh Summation. And from 2 to Infinity this is going to be eggs to the power. Excellent power, naked and -1. Whenever I multiplied by this one, when I multiplied by this one, then it's gonna be plus summation. And from two to infinity An and -1 exit pile again. Now I'm going to combine the summations. So this is gonna be after ffx summation and from zero to infinity uh and multiplied by and minus one. Multiplied by, you know, Exit power in -1 And -1 plus exit power here. That is what I have. So this is the para serious representation. We can try to make it a lot nicer because whenever we want to find the radius of proof regions, we're gonna struggle with this one. So you can make it a little more compact. Let me see this thing, 1/1 -1. When you take two derivative of this one is going to be two or one of my inner space. Cute. So there's supposed to be a two here. So in order to get rid of that too, I'm supposed to divide To multiply here by one half, so one half, one half and that is going to cancel this too, so is going to affect things uh Here is gonna be one half, here is gonna be 1/2 Okay, okay, so uh you know, I still want to try, I'm still trying to find a way to make this thing more compact so that I can do the ratio test and I think I might just, I might have just found a way uh Okay, let me take this one off. Mhm. So you know this thing here is going to be this too, I can bring it here, so it's just gonna be uh summation that Okay, extra power in -1. Big problem then over to, I just brought those two here, I'm going to bring this to here as well, so it's going to be plus summation and from two to infinity and -1 uh exited power in over two. Again, what is somebody going to do If I put em plus one word I see in here, it's going to allow me another, ready to change this summation to start from one to infinity wherever I see. And I'm gonna put in plus one, so this is going to be in plus one. If I put em plus one here, you can see that, I'm just gonna have and if I put in plus one here, I'm going to have exit the power and and Justin then over to Okay, So you put into this one here and they change this 1-1 right? Uh, trying to make it a little more concise than plus. Now look at this one, okay, this guy here is the same as submission and starting from one to infinity of the same thing I'm gonna explain in a couple of minutes, a couple of seconds. Sorry, do you know why? Because whenever and is one If you put one here when N is one, this is gonna be one, This is going to be one And 1 -1 is zero. So it's going to make everything zero. So starting from one does not change this one, it's still the same thing because he started from 10. But I particularly wanted to start from one because I want to make it look like this one. What now? I can do some combinations. So when I combined you can see that I can pull out the submission. What else can I pull out? I can pull out an end because the end is common. I could I could pull out a 1/2 Because 1/2 is common. So I'm pulling out and then over to I can also actually put up pull out an X. To the power end. Because there is so common So that I have is n. Plus one Plus and -1. That's all I have left. I have this guy and this guy left so I just add them and you can see that this is going to take away that right So if that takes it away I just have twice of this. So that is a pretty nifty where you're putting it because it's gonna help you a ton with the ratio test. If you put if you left it in the way that I did in the first time you're gonna struggle to do the ratio test because it's actually gonna be crazy it's going to get out of hand. So this is gonna be Times 2 to the power to. And because this is this is taking away this and this is adding this one to give you two. And and actually this too is canceling this too. So it is getting way simpler by the minute and this and is multiplying this. And so finally you can see that in a concise manner. This thing is in squared X squared no X. And in a concise man. So I've been able to manage to reduce the series to this floor which actually works in my favor. So this is the power of serious representation. And this is why it works in my favorite. Look at a ratio test. The ratio test is looking for a influence one or eight N Less than one limit and approaches infinity. It is very easy to construct the sequence in plus one in the end. So what is a N. Plus one? It means I would probably see in here. I'm gonna put end plus one that is N. Plus one squared X. To the power and plus one that is my and plus one. Okay, so this is going to be Excellent power and plus one. And what is my hand? My hand is just the same thing because and I wrap some absolute value around it and put a limit and approaches infinity here and let it less than one. That's my ratio test. So you see how uh huh simpler it has become put in this uh series into this form. That's a lot of heavy lifting for you. Otherwise you're gonna stroke you're gonna you're just gonna struggle in that gigantic sequence so I can do something a little bit. What can I do? I can um This is going to cancel this of course. So you just have an X. Year. So this thing is limit and approaches infinity of n plus one squared over and squared. And how our next year less than one. Because this is this is going to be left out is the X. Uh logic do taste that I should uh you know expand this breath disease which I'm gonna do n square plus and squared plus two. N plus one over N squared X less than one limit. And approaches infinity. Known a divide both sides by N squared, divide every term this term that that and that when I divide each and every term of this uh fraction buy in square to have limits And approaches infinity of one plus to over end plus one over N squared Over one times my eggs less than one. So whenever I let uh you know and gold infinity this is gonna go to zero. It is going to go to zero. So I just have one and multiplied by X is just my ex, less than one. So my radius of convergence is just one. So that is the essence of making the series as concise as possible.