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(F1)"-1 b) Determine at least how Iany terms of' the series should be added up to app- (4n 3) (2n - 1)! n= sin 7 roximate the value of dr with a CTTOT les...

Question

(F1)"-1 b) Determine at least how Iany terms of' the series should be added up to app- (4n 3) (2n - 1)! n= sin 7 roximate the value of dr with a CTTOT less than T2 2021

(F1)"-1 b) Determine at least how Iany terms of' the series should be added up to app- (4n 3) (2n - 1)! n= sin 7 roximate the value of dr with a CTTOT less than T2 2021



Answers

The terms of a series are defined recursively by the equations

$ a_1 = 2 $ $ a_{n+1} =
\frac {5n + 1}{4n + 3} a_n $

Determine whether $ \sum a_n $ converges or diverges.

In this problem I can write the value of a person is equal to fight any square pledge to. When I can also write, the value of P. Two is equal to as to minus as one which is equal to five More duplication. Fool Plus two, multiplication to -5 Multiplication one. There's two multiplication 1. Which in solving further I get the value of the, so simplifying it further, I can write evaluate you 20 plus four -5 Plus two which is equal to 24 -7. which on solving further I get devaluate 17 who? According to the option A option B is correct. Option B is got, it ends it.

Series An equal to 1 to infinity of five and plus two upon two times and Squire plus three times and plus seven. So here in discussion, we have to determine whether the threes converges. So here we know that since the convergence is determined by the behavior of term for large end. So here we observe there five n plus two upon two times and Squire plus three in plus seven we have like five A bond to win as and approaches infinity. So we know that threes An equal to 1 to Infinity of five are born do and diverges because we know that in P series one upon and to the power p where B Greater than the one. So three's will be converses where P less than equal to one, so three's will be diverges. So here you can see that the value of P is one Because and to the power one. So here p equal to one. So we can say suites will be diverges. And we know that so threes unequal to one to infinity off five and plus two upon two times and Squire plus three times and plus seven diverges. So it is all finally.

You know this Siri's but is 1/3 minus one Hop waas one my life, Mine's one or, um, last one seven. My was one day. So all these will be It's obstruction off. Harris on the farm is something for one there is block term over theirs, long. Oh, yeah. So old is gonna be alternating Alternating on the Thames are gonna be like these. My nose. Come on. So, uh, or the Seine goes on Juan do breathe is or was a serious now, and they did it Suits. So So the rolling that, uh, it doesn't feed with very dear in or we'll stems are all was It was alternating positive terms was he did what they're hated over distance are not are not increasing. So this means. But for some there exists some you can So that, uh, in his beard on this capital and the term with this current that serious us Uh um some from I it waas one upto ability off the terms a seven. Uh, the term in seven plus one will be smaller. John, also you, as you can see here. Uh, well, this doesn't happen because, uh, one or 27. Uh, so I decided on being on over the side. You would have won or eight cities would be they want to for five term Sit. If there this is that a terrible. So, as you can see, this term, the eighth time is B. This is more is bigger by Misty's. Uh, she was six. So these Children, so the terms are are not, uh is not Don't be crazy or the Christian. However, since, uh, this serious have you written as, uh, be written us the following some some for my balls. Um, one when I go any day, something it I minus it is like, uh, it was like but So, uh, all these, uh, these terms since Jim these therms is absolutely convergent. But that means that not some of value in some of these Well, just the positive terms. That volume 33 I with some of the it's everybody. Oh, Bob, that's in the body of peace. Just getting rhythm with minors. This sum converges breads because, uh, said geometric you Is this obstruction off geometric serious. So, uh oh, for a jumper. Serious, Jim Magic. Serious issues over form one over our I for my Valls. One of the divinity. This is a job, Patrick. Serious on the some will, people toe. Why my nose? 1/1, minus R minus one. So this series is about these. The discomfort is as long as the absolute by your bar. This, uh, smaller and I want here. Uh oh. That somebody up there. This is what I want. What's a bio? One hop. This is more of one. So that the seriousness be a community. Arrange as the obstruction off geometric Siri's. These were mindless, but one on Dan, huh? We can Tuvalu worried it. We can use the formula. She's like this. So what? These only would be 1/1, minus one. There about my news wine. Uh, my, US, um one minus one over 11 minus one. Top minus one. Um, So when we're doing this obstruction, uh, the spiders. Why will cancel with this discomfort? Positive. So that's good. What are days of the denominator? These one is? You are free. So these number, this whole number is not the one over. You might not want about you. So thirst the summer order. You want over? Um, so this would be to minors one, But it's one on one out that is one hot. So that, uh, all these numbers, you will too. We cups on this number. Is he though? Just so. Over one or two. You will. So So these this whole serious. Okay, it's gonna be people to Riaz my school. So, uh, well, Jewish people to four house would be three minus or us people. Don't. We went for a nice minus one. Something this Siri's, I would remind us one, huh?

Already. Let's say we have a Siri's and we want to find the remainder of the upper bound. No. So we have the Siri's k equals one to infinity of a K where a K is equal to one over. Uh, okay, toothy eight. Reggie separates things. First, you're gonna have to use this nice little formula here, Uh, and this is gonna help us find the bounds upper and lower. So since we just want to worry about the remainder of upper part, which the, uh, persist, this is the remainder here. We're only gonna be worried about that. And essentially, we're only gonna be working over ring about this part here. So we're gonna want to find the integral as in at Infinity of one over Cue. The eight we can rewrite this size should be DK. We can rewrite this as, uh, key to the negative eight, but this is gonna equal negative 1/7 K to the seven and to infinity. You want this in terms of? So we're gonna want to plug these boys in, uh, minus negative 1/7 in the end, seven already. So this here is going to go to infinity. I mean, sorry to zero. Since it's one over infinity, and this is gonna turn to a positive, and this would just be seven in prison. So essentially, we have won over seven and to the seventh for our answer for that first part. Um, already? Yes. So that's gonna be the bound for the remainder or feel sorry. They're made for the upper bound in terms of end already. So now you want to know how many terms we need to make sure that are remainder here is gonna be less than, uh, 10 to the negative 30. So in order to do this, we're gonna have to re write this with inequality. So less than 10 to the negative three, we can find the reciprocal of both sides. What we need to make sure we change. There's, uh, in a quality. I went and use a calculator. So we're gonna divide seven to both sides or by both sides. And that is gonna be, uh, Mishka round. So 142.857 You're ready. And then we're gonna, uh, due to the power of 1/7 to both sides to get rid of this seven here we're gonna have in is greater than 2.316 And so you want tend to be larger than this. So we're just gonna go ahead and put and needs to be three, or I guess, and should be greater than or equal to, uh, three to ensure that our remainder here is going to be less thumb 10 to the negative already. So now you want to find her upper and lower bounds. So we're gonna go ahead and use this appear that I had this whole thing. Um, so this is gonna be our I guess. Like our guide, uh, to how we're gonna do our upper and lower bounds. So this year is gonna be our lower bounds. And this is gonna be your upper. So as n is gonna be our original, you know, uh, summation. Except it's gonna be from Cato, the one to the end. Since we have this and one over Kate of the eight Klis, go ahead and drop that down. And then we're also gonna have k equals one to the end, 1/8 plus. And already so now we can actually just leave this here, since we actually don't know what end is going to be so we can leave this as K a close one for the summation of K eight. And here, since we already technically did almost the same thing appear for the remainder, we're gonna replace n with n plus One. So we're gonna have won over seven and plus one to the seventh and then over here to the same thing. So summation hoops this week a, uh, Keiko's want to end, and then here we already know this is so it's seven in the end or times into this thing. So this is gonna be our lower and our upper bound care bounds already. And now we want to find ah, an integral We want to find an interval. Ah, which the value of our Siri's must lying if we're going to approximate by using the 1st 10 terms of the Siri's. So in this case, we're just gonna replace and with 10 and use these same He was here so we can go ahead and put 10. I don't know why. Write it like that one over K two b eight plus 1/7 times 11 to the seventh, and, uh, summation. Chemicals 1 to 10 but we're going to the K eight plus one. Time seven 1/7 times. Tend to this already. I used to calculate calculator for this. If you don't have, ah, calculator that the summations nicely. But luckily, these ones are relatively smaller numbers. So you just have to do 1/1 to the eighth one over to the eighth all way to 10 I. But if you do have a calculator, that'll, um, do summations. It's pretty easy. So we got one. We're sorry. The summation of chemicals. 1 to 10 of one of her. Kate is equal to 1.40773 and then 1/7 11 7th, which is this here is gonna equal. Ah, 7.3308 e to the negative nine. And the this over here 1/7 times 10 to the seventh is gonna equal to 1.42851 bus or seven one uh, each of the negative ready slivers gonna plug these back in. But what we have here, So we're gonna add these two together and then add these together. But he said then once we do that, we're gonna have 1.407735 a three and 1.407736 and this should be our interval.


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