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Problem IV (20 points) Using power series to solve the following Cauchy problemJ(? +l)1" + tr' +1 = 0 Uz(o) 2 (0) =1....

Question

Problem IV (20 points) Using power series to solve the following Cauchy problemJ(? +l)1" + tr' +1 = 0 Uz(o) 2 (0) =1.

Problem IV (20 points) Using power series to solve the following Cauchy problem J(? +l)1" + tr' +1 = 0 Uz(o) 2 (0) =1.



Answers

Determine whether the following series converge. Justify your answers. $$\sum_{j=1}^{\infty} \frac{5}{j^{2}+4}$$

Luminess is of the form one over chain terms and then about 10 of change jiggle from Judge Infinity and we can use in go desk here on now in integral task image computing the integral from zero to infinity No one over X Lanham about 10 off x dx And in this case, I can use the substitution. Where will you be, coach? You and ex. So you could you one of the X can chance believe it as well. So could you do Then you go to a land of Jew ex coach infinity. Then you could infinity. Therefore we can write is intercourse in terms of you. Now goes from an and a Jew to infinity And now we noticed an one of extra accent. Exactly a crocheted d'you Therefore we have that in the Kona becomes Indian. You are you about 10? Then we get this one, Nico, to the, uh your mom honest tending you. Then we get the Nuba minus 10. I minus nine. I even wrote it from Berlin of Georgia Infinity. So it we would infinite inside again is zero and we wonder and to gun. Uh, I learned up to about minus ni over night, I wasn't gonna be smarter than Infinity Neff By the intercourse test, we conclude that this recently we convergent

It is in the form phone chamber, 10 over chamber 11 plus one and jingles from one to infinity. In this question, the best choice for us two years will be the limit Racial test here. When we compare with us, he's under form. He had a maximum on the top of the chamber. 10. Inventing them by the Maxima in a part time job. Tripper. 11. Do we get equal to this so much in one of the tree and doesn't examine the harmonic series? Therefore, it would be divergent and that we need to compute limit on one of a J over for chamber. 10 over Jaber 11 plus one j. Guess to infinity. There we can simply fight and then we get the chamber. 11 plus one. Leave Alan by for Shane Bauer under 11. Here we can rewrite history, but it is one plus one over for, uh, it would be no one of a far less one of the four trade 11 Jacobs to Infinity, and it's Jenkins Infinity, this one with Juzo. Therefore, we have wanted to fall, and it's not make or do, and it's great and Enzo, therefore, by the limit racial test and this one and this divergent inner you conclude that this is here could be divergent

But it's a little too. Why crime us wise. He could stop. Okay, we know that white crime that's equal to the sum from energy quits one infinity of end times a and extra and minus one plus. Why, which is the sum from and as you could to girl to infinity of a end Sex extra pair of endless is equal to that. Okay, now the indexing. So this is the one we get. Just hold onto the same plus some from energy equal to one infinity of a N minus one extra and a minus one is equal to job. So this becomes a summation from N is equal to one infinity. It's factor out an expert opinion on this one. Times to end and plus hey, And like it's one we got to go. So we get that two and a n plus a n minus one is you don't r a n is equal to a n. What? It's one over two. Okay, so it's all for a one that's equal to one over to negative a knot. What is the two that's equal to negative one over 281 What a one waas a good one of virtues that we get negative one over to practice too. A knot a tree is a sequel to the one or 2 to 3 A. Not so you could see a powder from this. It seems like with this squaring or we're taking the power to of negative point over too. So our general form is why is equal to a not summation from n is equal to zero infinity one over for actually have one over to power and extra her oven.

In the home case grab us. One was square with a giggle Are one from zero to infinity. And here we can use the limit comparison test. Then we compare with issues on the form within a maximum power. Here. Do you have a guy squared over a squared with the came about five and no, we can simplify this recent Jenna for number one on a cable off half. So did these duck piece respond up English when, huh? And this morning, and one, it would be divergent. And I will teach computer limit. You know what number? Okay, about 1/2. All right. Case Graham. Bless one or more squared in a cable. Five plus one. Do again. Limit. Okay. Just your community you can with everything into a square root here that were heard. This one would be the graver five plus one. Oh, the this is gonna be a k rao five on. Then we will have here would be Ah, bless. Uh, this one will be better off. Uh uh. This will be the came off. Okay. Here and now here s gay is to infinity. We can push the limit inside a square root Then we will have this work because you two squared and, uh, limit came Mr Infinity. Then I'm sorry. Right? Everything by the table five. And then I should get the limit under one plus one of a paper. 51 plus one of a cable for. And then we will have his bigger to infinity isn't Here goes to zero Saddam serviced year zero. And then we have left with square to the one of the woman culture One because one speaker, then zero therefore but this divergent and battle in comparison test they concluded stories there will be divergent.


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