thus in this interrogate we ‘re looking at the taylor series problem. Let ‘s see the values we can get from the wonder. sol so far we have to find the LSE centered at a given value of A. Yeah is equal to pi and the function F effects is adequate to sin X. Okay, so we have to find the real disease. That ‘s the first thing. And the second gear thing is we have to find the radius of convergence. So lashkar-e-taiba ‘s starting signal by finding the dignities for this problem problem. therefore good to remind you, the format of the taylor series is um already out Ffx is equal to F of C plus F dash C X minus C Raise to Power one over one compliment. And this format over here just continues to carry on. So it will become F double smash at sea in two X -C. These two power to two factorial and this merely goes therefore on and thus forth. I ‘ll do another one to show you see ten minus C. Thank you the consequence on you. So this is the basic format of the taylor series. Okay, now let ‘s startle by finding these values over here. So we know that F of X is equal to Sine X. Let ‘s find the future force we can help identify the traffic pattern. therefore degree fahrenheit dash X will be equal to Cossacks will be equal to Cossacks. F double lash X will be equal to minus sine X. This is precisely dim-witted differentiation. We keep on differentiating to get the adjacent startle. F Q X will be equal to minus campaign X differentiate this. Again efforts to power four ten. The fomite to cynics there will be a continuing pattern that will go from this and forth. So F of five will again be cossacks then it will be minus seven X minus cossacks and indeed on and so forth. Okay nowadays let ‘s begin by plugging these values that we got with the A. So we know that E. S by. So lease ‘s write then F. Dash I mean F of pi Will give us a sign off by which is zero. then fluorine hyphen of pie will give us caused by which is -1. then F double dash private detective will return zero if thank you bye. Which is minus course by which will return one and finally have four of Which is F signed by which will return zero. So we noticed the depart in you. It ‘s 0 -101 zero. The storm which we calculated was Yeah 0 -1. 010. Let ‘s start by plugging these values in and see what we get. So the first term is FFC. FFC is nothing worth zero itself. thus zero plus F. crash of sea will be -1 and then we just write that out as it is x minus. And as we do n’t see spy bye Raise to Power one over one factorial. then we read the second term out which will now be zero. so if it ‘s zero it ‘s going to get multiplied by the moment term. So the storm becomes invalid. I ‘ll equitable write it out to show you zero into X -9 sq two factorial. Since this term has zero in it, it ‘s pleonastic. We do n’t need to write it because it will equitable cancel out in the future. So that ‘s the moment. now you get the other term after zero, we get another one again. So we put it one ah x minus pi Raise to power 3/3 factorial. OK, indeed now you ‘ve got a basic idea of how ah this serve is in serious how the following value comes. So lease ‘s identify the convention. We see that it ‘s one year and it ‘s three years. so immediately we can see that there is a common remainder of two. Same for this, it ‘s increasing by two. And what else has changed ? This first terminus from -1, it ‘s becoming one. It ‘s alternating. so now we can write the taylor series by identifying this pattern sign, X is peer to put the sigma sign and peer zero eternity. And let ‘s start by identifying the traffic pattern. indeed as we can see this minus one is keeps an alternating alternating subtraction 1212 subtraction 1 to 1. This traffic pattern will continue as we see him. therefore in order to get this down we can just write that vitamin a -1. Used to power and that way we can get the part in each board as and moves there. It will keep on alternating. immediately to get this we have to use charismatic progress. And the convention for that is we fair have read the first prison term which is a. Is equal to one And he is equal to two. This rule states E plus And -1 D. This will give us the next home in the seas, so is equal to one plus and minus one into two. So one plus two and minus two that will return To end -1. Okay, indeed nowadays let ‘s see the second separate this part, so it will be ten -9. So we want that issue, we equitable put two and minus one. So that direction we can get 13 therefore on and so forth in that traffic pattern increasing by two each time. And the denominator besides follows a like pattern with a remainder of two. So we can write that as uh huh two N -1 and the factorial symbol, yep. so now we have identified the fallacies in this equation and if you run the series will see that the first term will be zero. The irregular term will be this, the third term will be this and that ‘s how we get the hand is his. Okay, thus now we have completed the first partially of the interrogate to find the taylor series. Let ‘s look at the second character of the question and the second gear function of the question asks us to find the radius of overlap. now we have to remember when finding the radius of convergence of a series. We have to run the proportion screen. So let ‘s run the proportion test on this equation. Are there issues as states ? I ‘ll precisely write down with the regular states when restrict and tends to infinity That it will be a and plus one. Mhm. Just eight. And is equal to. now when L is less than one, we say it converges when L is greater than one, we say it diverges and when L is equal to one, this test is inconclusive. Okay now let ‘s run the issue test on this. Uh huh. I ‘ll be right. There comes out it will be -1 and plus one and two X minus pi two and plus one minus one over two and plus one minus one factorial. And this again will be into because it ‘s division. When you ‘re dividing we take the reciprocal of the denominator and multiplied. So there is a broadcast that will be, it ‘s equitable this equation, this one but the rest of advance. so taking the reciprocal of that, we can write two and -1 factorial over minus one, arouse to office and adam minus by two and -1. Okay let ‘s equitable try that a bit more clearly and make it a bit easier for us to solve. then this on the left bridge player side we can reduce this, we can write this as minus one, Press the world power goal into -1, Raise to Power one into again ten minus five. Open that up. We ‘ll get two N plus two minus one. That will give us two N Plus to -1. That ‘s one. Okay now let ‘s look at the Dinner director, Robin until we get millimeter two N plus two minus one. That will give us two N plus one factorial and it ‘s being multiplied by again two and -1 factorial over minus one and x minus private detective two and -1. so this storm gets canceled with this as they are the lapp. This gets canceled with this, is there the same ? And finally we can observe that one plus one. All seems to be one plus one. Yeah all teams should bring well sol far and ultimately we can observe that We have the final examination equation as -1, Raise to Power one into two and -1 factorial and over two and plus one factorial. So this is the equality that we obtained from running the the proportion test. Yeah. thus now we can try and simplify this a bit for them. There ‘s there ‘s no degree in writing anything race to power one, is that act itself, yep. And um now we can precisely reduce this equality. so by reducing this equation we can get the follow, yep. so introducing this equation we will obtain Yeah, so that ‘s the interrogate. I ‘m good-for-nothing for the delay, I will be -1 and two. two N -1 factorial. And now let ‘s start opening the bed equation. It will be two n Plus one. That would be the foremost time. Opening up that exclamation find. then it ‘ll be two N plus zero. Which would barely be to end and then it ‘ll be two and minus one victoria. So this is gon na finally canceled up with this and we obtain the concluding term to be minus one over four and feather plus two. goal. Okay so now we know that N is adequate to eternity. so when we came eternity it will be one over eternity Which is equal to zero. And when it ‘s peer to zero as we said earlier that means L. Is less than one which means it converges. Yeah.