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3. How many terms of the series would you need to add to estimate the sum to n(lnn)? n=2 within 0.012 (Hint: YOu want Rn 0.01.)Estimate the maximum possible error w...

Question

3. How many terms of the series would you need to add to estimate the sum to n(lnn)? n=2 within 0.012 (Hint: YOu want Rn 0.01.)Estimate the maximum possible error when the 2Oth partial sum 73 is used to estimate n=the sum S = 24 n=[b) The 20th partial sum is Szo ~ 1.200867842__ Find a short interval (a,b) which contains S_

3. How many terms of the series would you need to add to estimate the sum to n(lnn)? n=2 within 0.012 (Hint: YOu want Rn 0.01.) Estimate the maximum possible error when the 2Oth partial sum 73 is used to estimate n= the sum S = 24 n=[ b) The 20th partial sum is Szo ~ 1.200867842__ Find a short interval (a,b) which contains S_



Answers

a. For the series $\sum\left(1 / n^{3}\right),$ use the inequalities in Equation $(2)$ with $n=10$ to find an interval containing the sum $S .$
b. As in Example $5,$ use the midpoint of the interval found in part (a) to approximate the sum of the series. What is the maximum error for your approximation?

It's under fall minus one hour. Okay. Over to gay plus one. Pretoria goes from George Infinity that we have any good. You three here we that 25 is only the ants and recon under pressure some s and gender, some from Sergio and Ali. And then to give this one fa toria. And by the room 10.18 you have the s a minus asked this absolute hair is Kandahar era and it's a Rohan will be smaller than a ecology. The next term I am risk one. And here were given the antiquity three. So what you need to replace the inhibit three And this one will be the A three here and then because of absolutely so we don't need, you can buy them. Understand? Okay. Therefore we have the one dividing by do Thames three but one Victorio, You go to one of ah seven Vittorio. There we get equal with you. Nothing. The one bond I ate for times 10 bermanis for

Okay. So we wouldn't find out how many terms of this series we have to add up to give any release, then 10 to the negative three. Um, And for that, we're gonna use the fact that we know the absolute body off there or is less than or equal, then, um, the first term that's being neglected. So since we want that their error to be less somethinto a negative three, we're gonna choose on in such that the first time that is neglected. He sliced somethingto a negative. So that's the expression that we want to achieve for that. Um, let's so foreign. So let's take exponential both sides Now, from the left side, we have e l n and those two, they, um, onto each other. So we get one glass, one of 11 plus one, and on the side. Uh, this number is one point. Ciro Ciro wants your serves 05 And so what we want is thes. We can't obstruct warm both sides, and we have that one over in. Plus, one is less than these. Now we can flip the fractions and calculate a number that is 999.5, Sirrah Sirrah, Sirrah, Sirrah, eight. And we need an end. The such that and blessed one is beginning about it. So it's enough to take in to be 999 And that's what can I do take in us my 99? Um, this should be 99. And so the some off the first nine 199 terms East seer Born 45 to 08 to seven. You can calculate these using actually also for so you don't have to add 999 terms by head.

Sulfur part, eh? Let's go ahead and find the partial. Some s ten. So that's the sum. Well, we start and at one stop it turn. So in this case, all the way up until and equals ten. And in that case, going toe a calculator, we could round off here. Such are approximation. And then we'd also like to say with the ear involved in in the approximation so here will use the formula. All right, so this will be our here here after using in terms. So in our case, using an equals ten, we'LL have negative one over three using the power rule. So just won over three thousand. That's the only heir that will have our upper bound for there. Let's go on to the next page for party. So this is where we'LL use Formula Three with n equals ten to him given improved estimate for the sun. So in this case, using three. Okay, So in our case, go ahead and plug in and equals ten. Now, we can evaluate these inter girls. We actually just evaluated this one among the bill. This one over here, just the choir rule again. So And also using the fact that we already computed this one on the previous page and got one over three thousand. So putting these together and the fact that we've already estimated this so recall this was about eight to, oh, to be seven amenable into a freezer Yet so now, plugging all this information in to the previous inequalities we have, we get the following plus one over three thousand nine hundred ninety three. That's a lower bound for the sum and then upper bound is plus one over three thousand. So just go to the next page and simplify that. Simplify those those sums. And now we have a better idea of an approximation for the sun. And you could even go ahead and just take the average of these two. So taking the average and there you could just round that off to about. So that will be our estimate for part B. Now for Percy, we'd like to go ahead and actually compare your answer and be with the exact value given an exercise thirty five, which ended up being kind of the fourth over ninety. That was the exact value of the incident. Some or in other words. This is the value of s. So in this case, we just want to look at the difference between kind of the four over ninety minus that our term up here from part B and going for a calculator. This is a very small number five zeroes after the decimal and then a foray. All right, there's one more part here. Let's go on to the next page. This will be a party. And here we like to find the value of end such that the following such that sn is within point zero zero zero one of the sun. So what this really means is and that should have been four zero. I'm sorry about that. Four zeros in that one. So me come back here and fix this. So now we have the following are in and to infinity affects the X. Now we can just evaluate the cinder girl. It's improper, but it's it's still doable. Use the power rule. And when we plug infinity, you get zero. So this ends up just being won over three and cute, and you want this to be less than zero point zero zero zero zero one. So go ahead and solve this for n So you need end to be larger than thirty two So you can go ahead and just take n to be thirty three or even anything on that we'LL be fine. So is Lana's. You're choosing any value and that's thirty three or larger then that will insure that s n is with them This number of the S.

But the Syrians under form minutes one about I'm the guy press one inviting like a about three plus one and then k goes from one to infinity. Here we are, given the in here Echo 23 and ah, recording by term 10.8 in here the ask and we just The bus was some here from 12 and minus one off guy plus one of any make able to press one and ever have the s and minus the asked literally with asked. Here the absolute hair is called the Aurora and it's a real here by former by term, it would be smaller than our eco Judah next home A and this one and they were given the and included three. Therefore, we have asked three here and then we have the dreamless one would be for here and now we have We continue to computer very under a far so we should get now for the absolute. So we don't need you can but a month one here anymore. Therefore, we should get equal to one Your budding by and China equal. Jafar also have a farm about dream this one and then we get wrinkled for balance. Three one and one here anyway. That and we get to go do one over the 65. I would be's upon 015 far.


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