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6) The radius of convergence for the geometric power series 2 n=0is R=s, (think about why thisis obvious) What is the radius of convergence for 2 ne0'where _i...

Question

6) The radius of convergence for the geometric power series 2 n=0is R=s, (think about why thisis obvious) What is the radius of convergence for 2 ne0'where _is Qy constant. Explain

6) The radius of convergence for the geometric power series 2 n=0 is R=s, (think about why this is obvious) What is the radius of convergence for 2 ne0' where _ is Qy constant. Explain



Answers

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {n^2x^n}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot \cdot (2n)} $

There were even does submission off the and it's grand experiment and dividing by June times, foreign times opportunity to and from one to infinity. And we need to perform the racial test now and interracial tents. We need to compute the limit over a in here and now we see that from the endless one begin the endless one square expert and this one off the two times phone times after the two and and then attempts to end this too. That's when we know I end this one. Now we divide by I and so we need to multiply by the reciprocal So two times, four times after that, you and Armando and squared times expert. And now we say we can cancel all of this one with this form expert and with this one, and then we can good off access outside and we will have the inside becomes the absolute of xperia times the limit. Oh, no, we're having that, uh, and this one Jew dividing lined up and squared attempts to endless to and because the degree under denominator history that between the numerator equal Cheechoo only there for the whole limit. Here we're goes to, is there? Ah, And then we get in culture. I'm sort of thanks Times their own it could you Zero and zero is smaller than one. And this one in means done That series here will be convergent Ah, for all X in real number and isn't in less than the interval. Gojira Romans in vintage infinity and the radios go to infinity.

Okay for this problem we'LL use the ratio test first. Just figure out where we get convergence. So Lim has n goes to infinity of a and plus one over a n and I and we mean this whole chunk here, including the X values This is X to the two times in plus one over and plus one Sheldon of n Plus one squared. So that's just our A And plus one term we're dividing by a m which is multiplying by the reciprocal the van. So what now We're multiplying by n times natural log in squared and we're dividing by X to the two n so x to the two times implicit One is the same thing as X to the two n multiplied by X squared so extra too And we'Ll cancel out with this x to the two ends and we'Ll just have x squared there and then here we have an end here we have an n plus one Well, group those terms together and then here This is an exponent of two This is an exponent of two. Well, also lump these terms together as n goes to infinity Natural log of and divided by natural law. Govind plus one is going to go toe one. You can see that by applying low Patel's rule and as n goes to infinity and over in plus one is also going toe goto one. So this is just going to be absolutely You have X squared and we want for that to be less than one. Okay, but X squared is already going to be something that's non negative. So x squared. The only real condition here is that X squared is less than one. Okay. And this means that axe is going to be between minus one and one. So we get that by taking the square root of both sides square root, and then doing the plus minus. And you get that exit between minus squared of one and positive squared of one which is just minus one and one. Okay, so our interval of convergence is somewhere from minus one. No one. At this point, we don't know whether or not we include minus one and whether or not we include one, the length of this interval is going to be to go, which means that the radius of convergences too divided by two, which is one so one is our radius of convergence. To figure out the interval of convergence, we need to know what happened when we plug in minus one into here. And what happened when we played in one into here? Okay, but minus one to the two n is just going to be one, because the exponents going to be even and one to any power is going to be one. So we're going to get the same case when X is equal to minus one. That'LL be the same thing as when X is equal to one. So there's really only one case we need to check here. Okay, So when excited, too. And get your place with one. We need to ask ourselves what's happening then We have one over in time's natural log in squared, and here we can use the integral tests to figure out whether or not we get convergence. Okay, so figure out whether or not this integral is convergent or not. Case will be where change of variables here. So we'LL do u equals natural log of n which means that d'You is one over in d n A. So whenever we see a one over Indian and this equation will replace that buy, do you? Okay. So if n is equal to two, then you is equal to natural log of two. When n is equal to infinity, you is equal to natural log of infinity, which is still infinity. Here we have a one over and Deon So we replace that buy, do you? And then here we have natural log of end but natural lot of vintage issue. So here we just have one over you squared one over you squared is just you to the minus. Two of you write it like that. It might be more clear how you evaluate this. We just do the power rule for inter girls. So this is going to turn out to be minus one over you evaluated from natural log of two up to infinity. Can we like to think about this is a limit, So All right, Lim eyes, We use him. Hear his m goes to infinity minus one over you from natural log of two, two em. So that's limit. His m goes to infinity one over. Sorry, minus one over. M minus minus one over. Natural log of two. This term is going to go to zero. Here we have negative. Negative. So that's going to turn out to be something positive. But the important part here is that this is this is something that's finite. So the interval is going to converge. So by the interval test, we get that the some that were considering is also going to converge. Okay, so both of these in points are going to be included. So our interval of convergence, we include minus one, and we include one.

So define the radius of convergence for pa series. So you're looking at radios of a convergence convergence convergence of a power series that was seems. So what we are saying that if you pick the sets, so yes, he states of uh of our values of X for which we are saying that the power series, the power series is convergence or the power series. Okay, so for which the use of X lucy Sets of all values of eggs, then we are seeing one, the power series for which the power series for which the power series the power series convergence converges to the interval convergence convergence is the interval over is a endeavor? Is the interval of convergence of convergence? Okay, So if the power of the past series converge for all is so if the power series, so if so if the power series, the power series converge converge for or X then r radius of convergence, then our radius videos of convergence convergence is going to be oh equal to infinity. Are you calling to infinity? But if the power series, so if the power series, the power series converge convey gis it's only see then are it's because zero. Otherwise, you remember in a study of theory um and our power series, We had theory 9.0. Is this that there exists? He says there exists um a real number there. Is this a real, a real number? A real number? Uh huh. Greater than zero in this hour. Okay. Is all we are saying is what's the radius of convergence? Videos of convergence. Okay, so because I'm using the videos then let's use they are to be this uh Okay. Is the regions of convergence such deaths, such debts? The series, the series converges absolutely, absolutely. Four. So you have x minus C to be less than our then diverges. So diverges diverges for you have x minus C to be greater than uh But its converges, the series converges absolutely. For all X that is under theory 9.0 under power series. So with this and yeah, we are just defining the videos of convergence of a power series. So for a series censored at sea, so there are series centered at towards D. C. Then the following we are stating is so true.

Emergence power series. Use the ratio test. So you have extra two and plus one, which is just two N plus two. Same for 2, 1 plus two. Factorial. So it's going to be multiplied by two. And factorial Oliver next to the two and kept. So what happens when we simplify this and take it to? Well, let's just take it one step here. Which is that if we simplify this down, we have X squared on top because of the X to the two end leaves us with this X squared here. But then in the bottom right, the two N. Factorial cancels out with This which expands out to to end plus two. It's 2, 1 plus one. It's too And factorial here. So these cancel But that leaves us with to endless too, times two and plus one here. But now as And approaches infinity here, this goes to zero, which is always going to be true for to be less than one. And so therefore are radius of compressions. His infinity because it's convergent everywhere from negative infinity to infinity. So that's our radius here.


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