Use power series operations to find the Taylor series at x = 0 for x tan x5What is the general expression for the nth term for the Taylor series at x = 0 for x"...

Use power series operations to find the Taylor series at $x=0$ for the functions. $$x \tan ^{-1} x^{2}$$

Let's start by simply thing what we have because you know an expansion for sine X. We know an expensive of course, I next, but multiplying these two, it's very complicated. So let's use to streamline our identity. So we know that two times sign X Times Those annexes signed two x So that means sign next time school sine X slips dominated period, So sign X Times course annex. It's just half off. Signed two weeks. There is word because we know an expansion for sine X, and hence we know an expression for signed to access will. So what? I mean this we know Sine X is given by X minus X cube over three factorial plus extra, the 5/5 factorial it minus extra the 7/7 factory and so on. So signed two x so half signed works is just given by half times about, uh, expansion with extra placed by two X because I decided next and we want signed two x so two X minus two X cube over three factorial plus to excuse the 5/5 factorial I'm so on U minus two extra the 7/7 factory and so on. Okay, Which you can simplify further as half Afghans to us. Okay, let's simplified properly. Half times, tourists one. So this is X 10 minus. Here you have to cube and excuse of two cubed times. Office to square Soto Square X cubed over three factorial. And then you have proved the five times half of this to the fore and the next to the Fife over five factorial minus your crew. The seven times half just from the six x to the seven or seven factorial and so on. Okay, now this is expensive for half signed poets, which is the same as I next time schools annex because off the trade identity that were ordered the top, this one. So we're done. So sign next time school Cenex is given by this expansion

We know an expansion for course annex and we know an expansion for sine X, so we can just Lundy's to win. So close I next is given by one minus X squared over two factorial plus extra 4/4 factorial minus extra 6/6 factorial and so on. Sine X is given by X minus X cube over three Factorial Etlis Extra the 5/5 factorial minus 67/7 factory and so on. So now let's open this second set off parenthesis and we can simplify from the first figured one and they knew it from the bottom of it. Minus times exodus minus six. Then from the top you get minus X squared over pu factorial From the bottom you get minus times minus x Cuba with three factory you just plus X cubed over three factory and from the top you get extra the 4/4 factorial and from the bottom you get minus extra the five over five factorial then from the top alert minus extra 6/6 factorial and then from the bottom you get minus times minus extra 7/7 Factory plus extra seven over San factory. I'm so on and this get goes on. Okay, So their pattern here is that the first Thomas one and then you have to negative terms than to a positive to negative into a positive and so on. The terms are ex expert over two factorial x Cuba with three factor land. So Okay, so this is the expansion for close an ex miner.

We know that eat the X is given by one plus X, but one fact told you plus X squared over. In fact, that's excusable. The three billion council so exited the excess, just the about power. Siri's multiplying by X, that is X times one plus excellent one Victoria plus X, where bless X cubed over three podium and so on on the way you multiplying. This is by multiplying X with each off these elements, so x Times one plus x times x over one plus x times x squared All the group put it and saw, and that leaves you X unless X squared over one factory in class. Excusable, you have to do plus extra for 35 billion. And so the next time index for the favor were for a missile. That's the publicity's for X idiots.

We know that on expansion for co Sign X is given by one minus X squared or two factorial plus extra 4/4 fact. Well, my intersex through the 6/6 factory time Someone Now what we need this X times cosign, Buddy X So what do you use to take about expression? You substitute by xto get co signed by X and then multiplying by X. So you get X signs the about policies but instead affects you use bikes so that one minus pi X square We're perfect close by X for the four before that, Putin minus phi X to the six over 6 500 so on so that simplify store extents one Linus by square x squared. What will that really plus by the forward extra before or for factorial minus spider six. Extra 6/6 factory and so long. And then now you multiply everything by X So all the powers increased by one. Say that X minus by square extra 3/2 5 total plus bite before extra 5/5 40 minus pi to the six extra seven over. Also, this is not five story but 4 15 It's only the power's off x increase because you're multiplying by X. So then by the six extra 7/6 factorial and so on, and that's the publicity's for exclusion by X.