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(20 Po nts)Determine the interval of convergence for the power series747(-3r)"+'In(l+) Find the Maclaurin Serics for f(r)...

Question

(20 Po nts)Determine the interval of convergence for the power series747(-3r)"+'In(l+) Find the Maclaurin Serics for f(r)

(20 Po nts) Determine the interval of convergence for the power series 747(-3r)"+' In(l+) Find the Maclaurin Serics for f(r)



Answers

Use Theorem 20 to find the series' interval of convergence and, within this interval, the sum of the series as a function of $x$. $$\sum_{n=0}^{\infty}(\ln x)^{n}$$

We wanna find the MacLaurin series for If it's equals a to the two Ed's we don't Want to Do is start with a Siri's that I know into the eggs. He is the series for K equals zero to infinity uh, one ever que factorial tons aids to the K Now turn by term. Let's write that out. If K were zero, you would have won over the road factorial times. It's to the zero. Then if they were one, you'll have 1/1 factorial tons. It's to the first UK we're too. We have won over two factorial times X squared Plus now Kay were three one over three factorial tens its keep, and this would continue. Now we want to write the Siris for a to the two eggs. So what that means is, everywhere there's an eggs were just going to replace it with two EDS So the song from K equals zero to infinity of one over Que factorial. But now, instead of Ed's runs to the K, we'll have to. It's rise to decay. And so, term by term, this would bay were I over zero factorial tons to eggs to the zero plus 1/1 factorial terms to eggs to the first plus one over two factorial terms to its squared, plus one over three factorial times to its cute, and that pattern would continue turn by turn. So now the one over Z zero factorial is one, and the two x to the zero is a one. So that first term is just one when you have plus. Now 1/1 factorial is born into its to the first would just be to its plus. Now 1/2 factorial is 1/2 and then the two squared is four. So notice four, divided by two, is two. So this one is too. It's squared. Plus now 1/3 factorial that's one sits and the two cubed is a So what you have is eight seats, and that will be 4/3. It's keep and the pattern would continue, Um, now above for each of the AIDS, I want you to notice that the 1/0 factorial was warned. Tons it's to the zero was born, so that was warn Plus And then we had heads and then plus we had 1/2 it's squared and then we could continue simplifying term by term. Now the interval of convergence for either the eggs is negative. Infinity to infinity. So we'll use that to find Interval of Convergence free to the two eggs. So let me write that down. Wonderful. Ah, convergence for into the woods. Actually, I'm going to write it. It's evident Sequels, even the AIDS. So the interval of convergence for your F events equal t to the eggs is negative. Insanity to infinity. So let's think about either the two. It's now. So we're gonna find the interval. Let me write that a little better minutes that are did not look like an already did it. So we want to find the interval of Convergence four. And I'm just gonna call this one G since I called you to the itself. So for Gina the eggs Aiquile Teoh A to the two ed's Well, I know that the interval of convergence for either the eights is negative. Infinity to infinity. So what that tells May is that Nail MMA I'm going to take the two ends and it will be between negative infinity and infinity. Well, then, when we divide by two a number times Negative infinity. That is like negative 1/2 comes infinity. That's still negative. Infinity waas the nets less san infinity! So your interval of convergence is steel negative. Infinity to infinity.

Given, uh, Siri's under farm X squared minus one. I would Teoh totally about the And so it isn't arias. That doesn't give me the farm under G o Matic Siri's whether our hand in code you the X squared minus 20 when you make you so you need to have a key images three convergence even only if the X square minutes wanted very mature. Just absolute. Must be smarter than one and isn't every only if x squared minus one Anybody but you will be tremendous one and one and even only if x squared minus one when we between minus two and two and even only f X square must be be training three and minus one We know for sure than the X squared I always graded then minus one Certainly true on the time and for the other side. Here we have the X quest wanted and three even. Only f The axe must be between minus quick victory to square the three. So it doesnt implies that the radius under convergence coaches got a tree and the interval you go, Joe Romanus got a tree to grow the three and now does some by the gym. It is formula wishing and Ecco Joe the first thing, which is one. The finding by one minus the X squared minus one day of anybody to we can't seem to find is running again that you on the two minus X Square plus one. They were getting contractually of anybody three minus X squared and in its this some of the Siri's him in terms of the angst.

Number twenty. We have f X equals E to ex Montecito connected. That's over too. Okay, this one's interesting. So for eat with acts we know is want plus ax plus x squared over two factorial plus x cube over three factorial closed today, thanks to the ends over in factory. And it was an activist. We just substitute exponent u x one lands X plus x square to factorial plus So this one's going to be negative. X cube, three factory extra ends and factorial cluster So you can see this term and this when you plot with its sub tract. So this term cancels this term cancels. So the human power or Kansas to our power becomes of class. And did you want to buy two? So you got is actually two acts plus to multiply X cube free factorial plus candida to multiply acts to thee two and plus one I'm going to say two plus bond. Victoria, this guy has to be art because they're given turn Kansas dropped by to which he was X plus askyou through factorial applause. Taxi factorial Plus you can write exclusive to AM clasp on to Angelus on factory. Okay,

Stories goes from zero to infinity and inside have each of the X minus form Don't worry about. And so this one It is exactly the geometric series which is under form Some shunned. Uh uh, I times Allah and from research infinity. So this I, Herrick or Juwan on Ah, here every echo Judah a X minus far They're far by the Jill Matics Raise within. Get this serious convergence even only if ne e x minus farm, which is insane here. Absolute. It will be smaller than one. And from here we should get uh uh, Each of the X minus four must be between minus 1 to 1. And then we begin to each of the ANC's Must be between five and then this one Good be, uh, three. And to get the X with thunder Ellen everywhere So would begin NDA land off three smaller than angst and smaller. And then a five on From here we ganda writers under Convergence coach and end up five minutes, Alan a tree. They very much do so we can write its now as the half. I'm the and I'm off five out of three. That's where the Rangers under conversions and funded interview under Convergence, this will be the interval Echo Joe, You're some kind of trick to the end of five and phoned as some again. But you images sarees. There's some each of the x minus for about and it will echoed by family. We will be equal to the first term and with real borders are into the and regain the first time you go to one dividing by one minus Whatever inside here. So we have a X minus far and then we get equal to one over. We have here will be up three minutes each of the X, and that will be the some we're looking for here.


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