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Point) (a) Approximate the sum of the seriesby using the sum of the first terms_Answer:(b) Estimate the error involved in this approximation by using the remainder ...

Question

Point) (a) Approximate the sum of the seriesby using the sum of the first terms_Answer:(b) Estimate the error involved in this approximation by using the remainder estimate for the Integrab Test Answer: R4(c) What is the minimum number of terms required to ensure that the sum is accurate t0 within 0.04? Answer: n

point) (a) Approximate the sum of the series by using the sum of the first terms_ Answer: (b) Estimate the error involved in this approximation by using the remainder estimate for the Integrab Test Answer: R4 (c) What is the minimum number of terms required to ensure that the sum is accurate t0 within 0.04? Answer: n



Answers

(a) Use the sum of the first 10 terms to estimate the sum of
the series $\sum_{n=1}^{m} 1 / n^{2},$ How good is this estimate?
(b) Improve this estimate using $(3)$ with $n=10$ .
(c) Compare your estimate in part (b) with the exact value
given in Exercise 34 .
(d) Find a value of $n$ that will ensure that the error in the
approximation $s=s_{n}$ is less than $0.001 .$

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.

Our F. S. So E. Our if eggs which is equal to one divided by eggs to the powerful is positive, positive and continuous then F. Prime F. prime will be equal to -4 Divided by X. to the power five is negative negative for thanks. Great Saddens zero. So the integral test applied here. So this implies that's we have this in place that the sum And from 1 to Infinity of once the power and the one divided by answer the powerful it's equivalently s saying which is equal to Y. Singapore. one divided by four floods. one divided by two extra power for I for so I have one, one divided by once the powerful floods. One divided by two to the powerful Flights. one divided by three suit a par four plus to that same same which is one divided by saying see the powerful and this is equivalently 1.082 037 So are saying less than or greater than the integra. From 10 to Infinity of one divided by thanks to the powerful. The eggs will be equal to the limits as N approaches infinity You have one divided by -3 X. Cubed from saying to see. So this will be etc. As T approaches infinity. So this and it's equal to the limits. As he approaches infinity we have minutes. one divided by three C cube Plus one divided by do you read same? Q. And this is equal to one divided by 3000. So this implies that the arrow the error is at most 0.000 theory. Then for B. So for the second part E. You have s sane blazing Segre from living to infinity. One divided by X. To the power for the eggs less than or equal to X. Which is less than or equal to S. 10 plus PNC Girl of dads which is From 10 to infinity of one divided by X. To the power for the eggs. So this implies that our asking Plus one divided by three. You have living to the power three civilized cycle to S there's an equal to S. 10 plus one divided by three. You have seen through the power tv. So then this implies that we have 1.0 082 037 plus 0.00 0- 50. And this is equal to one point there it's 2287 less than or equal to waste less than or equal to this bus which is one 1.0 8-037 plots 0.000333. And this is equal to 1.0 82 82,370. So this in place that we get is to be equivalently one point zoo aids 233 With error less than Or equal to zero points 000 05 So what it means is that this estimates in fact be we are looking at So see the estimates in parts big. So estimates in part B B. Is you have your ass Which is equivalent. Sleep one when several 8 to three. Theory With error with error Less than or equal to 0.000 05 So the exact value. So this implies that the exact value, the exact value in exercise 35 is bye Superpower for divided by 19. Which is equivalently one. So it's 23 two theory. So the difference then the difference is less than 0.000 00 What? So then uh and R N. Less than or equal to. Then see girl from N to infinity. You have one divided by eggs to the powerful the eighth. And this is equals one divided by three. And cue. So this implies that are in It's less than zero points 00 001 Which implies that one divided by 3 to the power and cube Is less than one divide by saying to the Power five. Which implies that's three. And cue would be great sudden Thanks to the Power five. So I am it's going to be greater. Done the key routes of Approed to the par five Divided by three and this is equivalently 32.2. So that is that is for and Greater than 32 assad results.

Okay, So for part eight year, let's go ahead and use s ten to approximate or estimate the infinite some s And also we'LL also ask How good is the estimate? And so here by as ten of course we mean the end of the ten partial some. So the solution for part A So first will write out as ten. By definition, this is the sum of s just for the sum from one to ten. So in that case, just go ahead and write out the few terms here and add up a bunch of fractions. This would take quite a while here so one could go to the calculator, Miss round off here it's a one point five four nine seven six eight and that's that's an approximation to us. Now the answer, The second question, How good is the approximation? Well, if you look at exercise thirty four you'll see that they give the value for us for this sum here the infinite sum given by Oiler. So in our case, the error is the difference between the exact value in the approximation. So since we now have the exact value a plug that in and we'LL also play guitar approximation as ten, and in this case we're getting about point zero nine five two, which is less than one over ten. So it's not a bad approximation because it's less than one over ten, but we can do better by taking larger. And so it's not a bad estimation, considering we only use ten terms, but one over ten just might not be accurate up, depending on one's purposes. So let's go on to part B now for part B. This is world will actually go ahead and improve the estimate from party, eh? Using Inequalities three and plugging in and equals ten into three. So what's good and solve this so solution? So first, let's recall from party what we found. We found that s ten was about Okay, so now if you will get inequalities three it gives up or lower bounds through the exact value in terms of sn and these inner girls here. So notice the difference in the lower bounds of the intern girls ones and plus one the other one's the end. So here, because we're to use any equals ten, so that determines that. And if you recall from our Siri's and has won over and swear. And so that means FX should just be one of the explorer. So let's go ahead and plug all this in here. Okay? So then now we could actually just go ahead and integrate this we do have improper and liberals here, but these ones are not too bad, because when you plug an infinity, the expression become zero. So that would become s ten plus one over eleven, less than or equal to s less than or equal to as ten plus one over's him. So then now one would use the information from part, eh? And plug that in for the S tens here. So plug those in for as ten and then add the fraction and so going on to the next page. All right, this out. So there's s ten plus one over eleven and then s ten plus one over ten and then just add those together. And there we go now to find value for us here. So think of it this way. We have an interval here, a lower and upper bound. And all we know is that as you fall somewhere in between here. So instead of choosing either then points another approximation would just take the midpoint of these two the average and so here will take us to just be the average. Here. So is an approximation to s. And this becomes also we'd like to know the ear involved here, and so there's a few ways to do this. So this is our approximation for us and so sensitive in the middle, the air is just they have the length of the interval. And similarly, you can do this half if you like, but because we chose the mid point these two halves of the same length. So that's one way to write it. Or you could just take the entire honorable length and divided by two. So the entire in ruling it would be the right most end point minus the left most end point and divided by two. So this is Yeah, this is the length of the interval here. And if you go to the Congo later, you see that this is indeed less than the point zero zero five. So the ear is improved. Bye. Using in part B by using the midpoint s Now, let's go on to the next cage for party. This is where we'Ll compare the estimate in part B with the exact value that we mentioned in part A. But this is a coming for number thirty four. So this is in number exercise number thirty four. So solution. So from party, we had a approximation improved estimate for us. So let me write that down five to two to and we also saw what the ear bound was. So if you look at number thirty four as we mentioned, this gives the exact value dudes Oiler. So in this case, we see that the air given by using this value of s go to the calculator here zero zero zero to a and that's even that's also Weston the point zero zero five. So this shows that the answer we're getting from part B is getting very, very close to the exact answer here. Okay, so we have one more party here party. This will be the final part. This is where we'Ll find a value and you just need one value that'll ensure not the ear and the approximation is less than point zero zero one. Okay, so solution Well, we don't We don't want to use an equals ten anymore because that's not a good enough approximation. So instead, we're just we won't plug in a value for any it. But we'LL use this inequality for the air here. So for this is for the remainder when using in terms. So this will be one over X squared. That's R FX. All right, so it's pointing at that's remind us where the one over exports coming from. And we want this to be less than point zero one. This is what we want. And so we're gonna solve end to make sure we get what we want. So let's just go ahead and evaluate that in a girl there that'LL be a negative one over X from the power rule. And then if you plug in infinity, you get zero. If you plug in and you get one over end, simplify this and then we get an is bigger than one over point zero zero one, which is a thousand so any value of and larger than a thousand would work. So the smallest one that you could possible use is a thousand won any larger and would work, but here all they asked for is a value. So in that case, we could just use this value here and by what we just showed taking this value of and will ensure that the exact answer minus this value here, the one thousand and first partial. Some will indeed me less than point zero zero one, and so that resolves the problem.

Okay, we're going to approximate the some of the series by using the first three terms and then we are going to find the upper estimate. So notice if we put a one in that negative one is going to be squared, so we are going to have a positive first term And then 1/1 to the forest. So that is just the value of one. Now putting a two in will have a negative and then 1/2 to the fourth is 1/16 and then putting a three in, brings us back to plus and then 1/3 to the fourth. So that's 9 81. So if we then some that we get 0.9498 for our estimate. And then we can look at our error by doing the absolute value of the next term. So the next term would be putting a foreign So a 1/4 to the 4th. And that means our error is going to be less than that 1/4 to the fourth. And so our error will be less than 0.0039. It's a very small error already.


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