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Tutorial ExerciseFind the radius of convergence and interval of convergence of the series. 10 n = 0 n2 + 1...

Question

Tutorial ExerciseFind the radius of convergence and interval of convergence of the series. 10 n = 0 n2 + 1

Tutorial Exercise Find the radius of convergence and interval of convergence of the series. 10 n = 0 n2 + 1



Answers

Find the radius of convergence and interval of convergence of the series.
$$\sum_{n=1}^{\infty} \frac{10^{n} x^{n}}{n^{3}}$$

But, uh, Siri's under phone minus one Bauer and expel of your and plus one running Baiju and was one attire from zero to infinity. Hey, I need your blind racial task now. And in a racial test I need to compute the limit off the AM This one of I am and rest you infinity. And was Uganda limit off? I am just one. So because of the absolute value, so we don't need to worry about this man is one about And so we should get expert under two and last three over the Jew and last three tile on development, I answer. We need to multiply by the reciprocal. So we that you am just one entirely running by expert. Your and less one. And we see it would cancel out this with this one. And then we have now with the X Square here. If we can solve this one with this one and then we have left with the Jew and was three times gonna do and us too. And therefore the limit now will become the leading and most infinity absolutely x square. They writing by two and three times in the two and us too, and it's and goes to infinity here. This limit clearly in goes to zero, and yours is smaller than one honest may, therefore by the racial that's here mysteries Him will be absolutely convergent for own X in green number and it symbolized. In the interval we go Germany prominent infinity edge, infinity and the radius A good infinity.

In this question. You were even the stories under farm ex minister do about and, um and then square brusque one for the angles from Serge Infinity. And the first time in two years, the ratio test. And then really JICA build the limit off that I am the swan over I am and goes to infinity and then we get a call to limit off the here for the i n best one. I was again the X Minutes job under endless one the riding by n plus one square plus one And now we divide by and so should be the attempts with the reciprocal of high end, submit her and scram bliss one, do your writing by the X minus two And they were saying that we can consider the explicit majesty about and with this power here and I will bring this one. I am assigned the limit. Therefore, we have ever equal to absolute on the X minus two times the limit off the X squared plus one dividing by. And this one square this one and goes to infinity. Hey! And understand because they have a 73 in the denominator and numerator now from this limit. Here we go to one here. Then we will have the seven shin Ah, absolutely. Love the X minus two attempts One here and now, In order to have the convergence, this limit him was a smaller than one. And therefore doesn't it last them? The ex minister must be between minus one and one initially implies as well done. The axe must be be 33 and one. So the next time we need to test the to end phones on this interval here. So for the exit code to one, we will get industries under form submission here, that's a serious. And then we have exit to one and then we're gonna minus one bell and other end square bliss one from Serge Infinity. So Mysteries is convergent by the whole delicate series. And then we turned the limit after one over and square brisk one and just your infinity and they were getting Could you zero now for the days off the three X equal to three. And then we get this Reese. Now it becomes the submission of the one other and square bless one from the infinity. This again Answer convergent by the comparison test where have won over and square blessed one Miss Bonner than one over and Square. And we know that a submission. The one of n square. It will be convergent by the peace Siri's and because you chew created and one here and we conclude under interval I'm convergent it were equal to it goes from 1 to 3 for the one and three were both given is a convergent if we use the clothes into Over Here and the Raiders ICO to one.

Okay for this problem. Our radius of convergence are we have limited as n goes to infinity, absolute value of a N over and plus one. And this time that's going to end up looking like in plus one factorial divided by in factorial. Remember the factorial function, in fact or religious like one times two times, three times four times Thought that thought Although I have two times in, there's going to be lots of cancellations happening here. Everything will cancel except for in plus one up top. Okay, to the N plus one will stay there and this limit is clearly infinity. It's our radius of convergence is infinity and typically with the interval of convergence, you would wantto check to see if you know, plus or minus our worked. But in this case, it's ours, just infinity. So our interval of convergence is still just going to be everything all the way. Everything between minus infinity and infinity

Even the mission off the expert And over an Pretoria from the surge infinity Here the first time, we need to compute the pressure stands, camps and interested as we need to do the limit off the I am this one other I am and just infinity on then was we get I am. This one was again the extra and this one over the endless one for Toyota. And then we divide. But I am so we need to multiply by the reciprocal. So will be the and for Italian over the Expo. And they will say we can consider expel. And with this power and fraternal with this factorial here and then we have left with the limit on the absolute of X development. Aimless one and just to infinity here Noticed that s and goes to infinity. This haunting him. You just Jews there off the agenda Limited coaches zero And so his is arm. You're smarter than one for own volume. The X in real number therefore, mysteries here will be gonna every convergent for all X in room number l train blast down the interval You go, Joe. Prominent infinity, infinity and radius It could You are


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