So this seems seem, uh, first derivative. KUSA next second derivative negative sine x no third derivative. Negative. Go sign X fourth derivative. Get back to sign X. Right. So therefore, derivative, get back to the sine X and then it continues the pattern. Still going to go through this one again and continues as a pattern right now you have pi over two. This is zero. This is one zero, you know? So that is the same thing. Negative one zero. Right, So? So that is that is Ah, the pattern that you can see here. So what is happening? If you look at this one, you can see a pattern. You see? It zeroes once and negative ones. Right? So is Tomblin struggling? So is tabling Oh, so this is first derivative. Second derivative third derivatives, fourth derivative. Right? Uh, so this is the anti derivative, right? The the root of violated at that. This negative won t in, right? It is negative One city. Ah, Okay, so we have to split it because someone some some are zero. So that split it some other 07 Someone Not exactly. So okay for ah, for even powers for even powers off derivatives. Right? Uh, even powers. I think I made a little mistake. What is? Sign off pi over two. That is actually one. It is rather cosom power to that zero by the opposite. So if I put pirate to here, coz and power to is zero right, So here is supposed to be zero and ones and zeros and ones. So see you see here. So this is zero now what is negative? Sign Pirate tatoos. Negative one Negative co sign zero positive sign piled with you one. Okay? Yeah, that is actually what it is. It is Costa Empire, that is zero Khost Empire reduce zero and signed. Probably two is one. Right? So that is what you have. Okay, so for even powers off derivatives. Easy. This is at the zeroth derivative. This is it. A second derivative inside of fourth derivative. So for even powers off derivatives, you can see that the function is so for even powers. Does Iran even powers free? Even Paris? That function is negative one. It is troubling between wanted negative one. Right? Because here's one here is negative one here is one. So for even powers, it toggles between negative one in positive. And that is why you have a negative one to the power in right todo to depict that one and then for art powers. So there's a right odd for odd powers. It is zero. Right. Our powers. This one here is odd. Third derivative is odd, you know? So zeros for our powers. Good. So what is a Taylor series expansion then? Ah, so you know, we already have the, uh We've written the formula from the territories a lot of times, So if you follow the tutorials diligently, uh, you see, So when you put everything in there, you're gonna get a terrorism use off one minus. Uh, what have ah, 1/2 factorial, right? Uh, X men despise X minus pi over two squared I That is from the first derivative. Ah, So this is, uh, the zeroth derivative. Second derivative? Because for even powers, you have something for our powers. You have nothing because the our powers is zero. Right? And then you're gonna get, uh, because final four factorial X minus pi. Would you to the pound for right? You get any parent? You know, it is just this ones are regions for even powers of the derivative. Right? For our powers were getting zero for even powers were getting negative one and one. Right, So that's what is happening. That is why I just free discarded the own powers. And I'm just writing the even power because the out of powers or zero so is gonna continue front forever, right? For different, presentable future. And I mean, for the ants derivative for the 10th derivative, right, this is the even power and derivative, right? So for the anti derivative, we're gonna get, uh, negative one today in Is this one that I've written? And then if you look here, you see that this one is to you. This one is for the other one is gonna be six factorial. So it is increasing in the powers it is increasing in multiples of two. So this is going to in factorial and we just reading a, uh a, uh a pattern free. And this one is gonna be expires. Pi over two and the powers of increasing multiples of two. This is two. This is for the other one is going to six, right? So you can just write to in here. So there is There is a parent, and then is that it can continue for the foreseeable future. So or is gonna continue into infinity? That's what I'm trying to say. Eso Finally. Then it's gonna be summation and from zero toe an right and this is gonna be the parents. So you just gonna put that one there, right? For all in, Right? So that is gonna be over two and factorial X minus a pi over 2 to 2 in So once you have that. Ah, yes. So, ah, you can go on to find the radius of corporations or whatever means that a problem. So that is the answer. That is the Taylor series expansion.