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3. Find the interval of convergence for:(x + 2)k 3kShow all work Be sure to test the endpoints if the interval is not infinite....

Question

3. Find the interval of convergence for:(x + 2)k 3kShow all work Be sure to test the endpoints if the interval is not infinite.

3. Find the interval of convergence for: (x + 2)k 3k Show all work Be sure to test the endpoints if the interval is not infinite.



Answers

Interval and radius of convergence Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\Sigma\left(\frac{x}{3}\right)^{k}$$

This is going to coverage to So in order to do that here. Mhm. Take the ratio test. So we take the limit as and approaches infinity three times and plus one squared over Into the end. Plus one Next to the un plus one times. Maybe he's in the end over 3 to hands three times and squared next to the end. So when everything is all simplified here we get X over here absolutely there up less than one, which means absolutely of X is less than eat. So therefore we're gonna go from negative he to eat Okay, since we're going for negativity, basically we plug this in. So at negative heat we have three k squared times they get a point to decay and then at the we have three K squared. So with that in mind could see that both of these when we take the limits as an approaches infinity of three and squared, then it goes to infinity. Therefore it diverges at the endpoints. Okay. The right side would be because of depressions tests the left side because of the all training serious test. Try that out there. And so then that means that our internal convergences from negative E. T. Not including here negative

Harewood recorded and having ah so much in under expert and from the June infinity. And we could, you know, 1/1 minus X and will be valid for the absolute X monitor and one. So now I noticed that it would replace this X here by the X minus three. So we do into something here. X ministry. It would be the X minus three. And then we do. So now, which again? The new form. This submission X minus three. Well, and from the region infinity. And then it We could do one off one minus. Now we have the X minus three. Here, you can send you find you again equal to one. Over. Yeah, one minus X plus three. And then we get equal to one of ah, four minus X. And we voted for the absolute X ministries monitor And one Anderson, Meanest, an X minus three will be between the minus one and the one Oh, it means that X will be between the fall and Ah, this one will be three minutes. When would be too. So this will be the interval optical pertinence. And this one with a functioning we convert you

Well of conversions to do that. The ratio test. And so we're gonna have to and plus two factorial times X over 3 to the endless one. Mhm. Divided by two end victoria here And also divided by X over three to the end. All right. So simplifying this out here. Get to end plus two was two plus one. We cancel out the two in fact, Orioles And then it's going to leave us with times x over three. We take out these He says with X were 3 to the first and so then this has the limit as N approaches infinity goes to infinity here just greater than once. Therefore, the series purchase only at the center, Which is that X equals zero.

I was raised on the form. Expound to cable one over the tree about K minus one. He and understand it we modified is seriously a little weaken the same the one on the top for the bottom. We can write a script, the three. And now it will be the pound, uh, to K minus two Now and then we can turn it into, uh, some mission, everyone to put them into the X over square, the three to play this one. And then I need to terms with the scare it off. Ah, here I go on Duh squint of three. Ballot three. So we have to attempt this skirt of a three about three here. No reason why I do that. Because here we can recognize this one will be that t o Matic Siri's. So we don't need to use the racial test of protest and record of the Germans. Histories are about OK. It will be convergent even only if the absolute ass smaller than one. And therefore we can conclude that this is here. We we convergent even only if the X over square the three absolute value must be smaller than one. Even only if I'm to a lot of ex smaller than squid The tree and isn't implies that X must be between manuscript of three to scrape the tree so we can Ghenda radius you go to the square, the three here and interval we'll be from an escalated treaty was going the three, not including the to end point.


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