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7. Find the radius of convergence of the following power series kz2k: 4k 22k: (a) (6) (c) Ck= 474 3k 2k +3k + 5* + 6k k=I K= K=0...

Question

7. Find the radius of convergence of the following power series kz2k: 4k 22k: (a) (6) (c) Ck= 474 3k 2k +3k + 5* + 6k k=I K= K=0

7. Find the radius of convergence of the following power series kz2k: 4k 22k: (a) (6) (c) Ck= 474 3k 2k +3k + 5* + 6k k=I K= K=0



Answers

Find the radius of convergence and the interval of convergence of each power series. $$ \sum_{k=1}^{\infty} \frac{(-1)^{k}(x-5)^{k}}{k(k+1)} $$

Landry this of convergence for this power series. Let's go ahead and use the ratio. Test three to the K plus one, six minutes to through the cables. one older At this one, Victoria Times that by K. factorial over 3 to the K. Next minutes to to the K. It's a pen from here, taking the limit SK approaches infinity. When everything is simplified, we have three times X minus two over a K plus one and this will go to zero, which is less than one. So therefore the radius is equal to infinity. The interval of convergences from negative infinity to infinity.

It's an interval of conversions for this and use the Test for this one here. Take the best care, precious infinity care route. Yeah, absolute value here. Some ignore the negative ones of the cases were taking absolute value express three squared two to the two K. Rather over 19 K. Incident. That's going to be equal to the limit as N approaches infinity of X -3 squared over nine. She's fine because that's less than said that less than one multiplied by nine. We have X -3 squared Assistant nine between 6 min three. Less than three. So splitting that up here negative three. That's the next place. 3" assistant three. It also tells us that the radius is equal to three by the way. Okay adding three to both sides. We have zero less than X which is less than six. It's testing boundaries So that x equals zero. We have it's a negative on. Okay, negative three to the two K. Here I care. So this just gives us things you want to the K summation of urges And have two x equals six. Just gonna have a negative one to the K 3 to the two K overnight. Kay chicken. Just kiss us. It went to the k jostle temperatures. So that means our interval. It's going to be from through 26

Some conversions. Little courage is for these producers. Shoot tests. Women escape purchasing thing. I'm just here. What Fine. Yeah. Okay. Text plans to okay. Some simple minus the King's one for 1. Take this opportunity just here. Okay. Start with that. It's true. Lots of three in the bottom here. Therefore, experts to its last three tells us that three is right. Yeah, spending two sides negative one five. Let's test those countries. Examples. Native one. Yeah. K over. Okay, so stick walls submission of making a cake. She checked Rachel's X equals five. Mhm. Two. Mhm. Okay. It's just the solution carriage. Yeah. Therefore interval Russians.

It is questioning even three. Sometimes four minus one hour. Okay. And then x about three K over 27. Okay. And can go from zero to infinity. He noticed that we can greatest attitude. Uh, minus X power 3/27. Totally about off K from zero to infinity. Here we can identify this in his hand will be that your Matics raise with the form of so much under our about okay from George Infinity. So here we have this one. We meet our and this was a convert just even only if absolutely, must be smaller than one. So it doesn't. We required that we're going to convergence even only if the minus thanks. About three on 27. I'm so love. Only all must be smarter than one every only If the X power three I'm sort of a little It must be a smaller than 27 on design. I mean, it's that we have the x hair must be small and and, uh uh, this one could be three. And then from here, which we get And also because I'm still value here. So we won t o about it from here. Swell from ministry and they're far listen implies land radios equal to the tree and the interval You go, June from ministry to three, then open interval here.


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