compute-the-sum-and-the-limit-of-the-sum-as-n-rightarrow-infty-sum_i1n-frac1nleft4leftfrac2-inright2

Question

Compute the sum and the limit of the sum as $n \rightarrow \infty.$
$$\sum_{i=1}^{n} \frac{1}{n}\left[4\left(\frac{2 i}{n}\right)^{2}-\left(\frac{2 i}{n}\right)\right]$$

Amy Jiang · Accepted Answer

But this it was like by multiplying in this one over end, on multiplying out beat this power to So we have the summation of Isaac to one and of we can get that as for and squared. And then, actually, that&#x27;s cute because it was squaring this. And then we have two scored, which is four. So we have four times four. So it&#x27;s not 16 and a nice word and then minus two over ence word I So I said this time here in this term here is a constant so we can put out on Let&#x27;s start by breaking this off into two summations. So we have 16 over ANC, you of I squared I Z one to Andi have minus two over and swear information from one on of I. So looking at, there are 2.1. This is Equation three, and this is equation too. Okay, so we&#x27;re rewriting them with their respective equation. This becomes 16 over an que challenge. I lost him and and plus one to end, plus one over six. And then we have minus two over end squared times and times on plus one. Okay, so let&#x27;s simplify of it. So we see that we have 16/6. Um, that&#x27;s equal to a two times three six of 8 8/3 two of eight. And then we cancel out one of these terms, and then we have to and squared. Plus bien, what&#x27;s one over three times? All right. Okay. And then we have minus. We can cancel one of the students, and then we have minus two and plus two over, and okay, now let&#x27;s combine this into a single fractions. Almost four, this time by three. Okay, So its walls, what out? Eight year. So eight times two. That&#x27;s equal to 16 and squared. Plus three times. It&#x27;s if you go to 24 and eight over three and squared. And then, well, two of minus three terms to which is six and squared minus six. And okay, so let&#x27;s simplify this. So this 16 this is 16 minus six. So we get 10 and sway and in 24 minus six, that&#x27;s plus 18 and plus eight over to be and squared. OK, so this is our fall information on. Now let&#x27;s find the limited. This so knows that we have our hash power is and squared, so we&#x27;ll divide by and squared in our numerator and denominator. We have the limit as un approaches. Infinity of 10 plus 18 over an was a spillover and squared over three to these council, and we&#x27;re left with a 10 open three for our limits.

Amy Jiang · Answer

So let&#x27;s every multiplying in this one over and And this getting rid of this power here so we can rewrite this as a summation of Isaac to one of two square That&#x27;s four I squared over. We have any times and squared, so that&#x27;s cute. And then we have plus four over end squared. Okay, now let&#x27;s break this up into Susan Nations. So know this year that for an end are constants. So we can just put out outside okay, and then looking at the other 2.1. And this is Equation three. And this is equation too. So be reading them in terms of our equations we have for over and cute. And then we have, um, in the times and plus one times two n plus one over six. And then we have four over and sward and times and plus one well lets you. Okay, so for over six, that&#x27;s 2/3 to have two or in councils, and we&#x27;re left with two and squared plus three n plus one over £6 squared that you have three. It&#x27;s weird now and then we have plus this end cancels and we have to and plus two over. And now let&#x27;s multiply this by end squared. Brush it out. It&#x27;s word, um, three and over three end to come by this into a single fraction. Okay, so multiplying out to we get four and squared plus 12 and plus one, but 600 plus six. Actually, this is an squared and over three and squared. And in combining the same terms, we have 10 and squared, plus six plus 12. That&#x27;s 18 and plus 1/3 and sweat. Okay, so we get the full information or some analysis, find limits as an approaches infinity, some of us that we have an inspired. That&#x27;s a harsh terms. So what&#x27;s the white but at so we see that these two terms cancel and we&#x27;re left with, um our limit is equal to 10/3

Amy Jiang · Answer

I just really want to find the full information. So let&#x27;s start by Rick News up into two creation. So we have one over N that I over end luxury ice on us. I squared over ends word and we have plus decimation of one over n squared. So what? It&#x27;s here that this n values a constant so we can just factor that out. Three at one over and you summation from ice equal to one to end of I squared across one over and squared Summation of one from eyes, people to one toe. So now let&#x27;s use equation in three. And this is equation one. I&#x27;m doing a 2.1 said if you have this one over and Cube times are last time, which is on times and plus one sometimes to end plus one over six and then we have plus one over and squared times. I lost him in an entire to constantly Okay, just simplify. Yes. Yet end with one to end close one over and squared times and plus one over end. Okay, so this is our station. And now we want to find the limits as and approaches. Oh, Was it approaches Infinity clothes, a multiply out earning writer here. So we have to and squared plus green on an influx one over until about two. Time six and n plus one over. And so from Mr here we can divide by our our high stone threats equal to the limits as an approaches infinity off two plus three over end, plus one over and squared over six. And then we have plus the limits as on approaches infinity of one over at I would actually fruit this time right here. Let&#x27;s combine this into distraction So we multiply this term by Yeah, let&#x27;s get out first before we actually divide by our house Far who will have to you any squared but three and plus one An inquest will multiply this by six on on both sides. So we get six n over 600 Skway. So this is equity six plus three just x 789 in. Okay, I know we&#x27;ll divide by our highest powers. That&#x27;s and quit. We get the limit as an approaches Infinity of two plus nine over end, post one over and squared over six. So taking our limit These cancel these becomes Errol Under left with our limits is equal to 2/6 and that&#x27;s equal to 1/3

Amy Jiang · Answer

Okay, so we want to start by solving decimation. So notice that we have a one over in here, and this, in turn here, is a constant. So we can actually this pull that out. So let&#x27;s start by doing that. So we have the summation, um, are easy with one toe end. Well, weaken, break up this I over and squared as I squared over and squared. So we actually have one over and cute. So yeah, let me to smooth this of it, actually. Zestful here. No, this. So we have one over and cute, and then we have minus will. Five is a constant. And then we can pull out that one over and as well. So we have five over in from Isaac with one to end of our I turn. Okay. And now, looking at their 2.1, we can use Equation three here and this. You can use equation too. So what does that give us? We yet one over and cubes are, or last time, which is 10 times and plus one times to end, plus 1/6. And then we have minus five over end times and times and plus 1/2. Okay, so it&#x27;s simplified. It&#x27;s a bit. So we yet that one of these ends cancel. So we have then two and squared. Plus, um, let&#x27;s see what is an equal to if I multiply that out So I get you in squared plus, um, and plus to answer. That&#x27;s three n and n plus one. This is over six pence word. And then we also have minus five times and squared plus a pen. Or actually, we can cancel out that end, right? She&#x27;s to terms here, so I just get five and plus five over to Okay, I know all multiply this term satisfaction by end squared. Actually three m squared over three and squared so I can combine this into a single fraction. So that gives us two and squared plus three and plus one over six and squared minus five times through. That&#x27;s 15 and cubed. And then we have, um, minus five. Were actually that&#x27;s 15 bartender three and squared over 6000 square. Okay. And then combining this interesting a fraction. We get negative. 15 and cute, uh, negative 15 plus two. So it&#x27;s negative. 13 and squared, plus B n plus 1/6 and squared. OK, so we get the full information on now, let&#x27;s find the limits as an approach ability. So in order to do that, we can divide both or divider faction by Attash Fire. So that&#x27;s, um, by and to the power of three. So we get negative 15 minus 13 over and was three over and squared, plus one over and cute. And this is over, um, 16 over. Yes. Actually, it&#x27;s not that we can notice that, um, or numerator has a hyper than are the nominees. So we know that the limits of the falling let&#x27;s rewrite this. This is equal to since our hearts powerhouses Negative sign. This is equal to negative infinity.

Doruk Isik · Answer

and this problem, we&#x27;re given a series and using the hint given a problem we could order given. Siri&#x27;s into this. Um, we&#x27;ll be fine, Sue. So in exercise 109 It&#x27;s given that the fever General Room a n minus two impossible plus at M plus two. And if as limits and ghost. Infinity Um, nothing. Zero. Then the summation off this series and then we&#x27;ll be f minus two. So we&#x27;re gonna use this. What if I series? And then we&#x27;re gonna use again the stick for exercise? Let&#x27;s define our life in as one over two, since that&#x27;s what? Because, as our initial eyes are person Oh, there we can right warrant over in plus one as two times one over two times in four and only two times one plus one and we can write one or two plus two as off plus two. Right. So by combining these, we can write the Given Siri&#x27;s s effort minus two have impossible plus plus, dude. Right? And if this r n by using, uh, what&#x27;s driving a juicy one of mine? Because this Siri&#x27;s done will be done at war sort of summation will be cooked by phone or any sector. Uh, what? That one is the people. Oh, we wanted to be one health. And when any student we find that Thio sport czar and right, it means that estimation, there will be one.

Fuzail Shakir · Answer

Okay, so this is the integral from 0 to 2 off one over one plus X squared DX.

Problem

Compute the sum and the limit of the sum as $n \r…

Problem 21 Medium Difficulty

Compute the sum and the limit of the sum as $n \rightarrow \infty.$
$$\sum_{i=1}^{n} \frac{1}{n}\left[4\left(\frac{2 i}{n}\right)^{2}-\left(\frac{2 i}{n}\right)\right]$$

Answer

$$\frac{13}{3}$$

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$$\frac{13}{3}$$

Calculus: Early Transcendental Functions

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Robert T. Smith, Roland B. MintonCalculus: Early Transcendental Functions

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