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With error of magnitude less than 0.1, the approximate sum of the series 00 (-1)7+l un 1 is 3nSelect one: 0 a. 8 0 b 2 C. 1 d. 3...

Question

With error of magnitude less than 0.1, the approximate sum of the series 00 (-1)7+l un 1 is 3nSelect one: 0 a. 8 0 b 2 C. 1 d. 3

With error of magnitude less than 0.1, the approximate sum of the series 00 (-1)7+l un 1 is 3n Select one: 0 a. 8 0 b 2 C. 1 d. 3



Answers

In Exercises $41-46,$ (a) use Theorem 9.15 to determine the number of terms required to approximate the sum of the convergent series with an error of less than $0.001,$ and (b) use a graphing utility to approximate the sum of the series with an error of less than $0.001 .$
$$
\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n !}=\frac{1}{e}
$$

In discussion here, Would you conduct the ask and minus the end in the Internet in stories here it would be smaller, decode and absolutely next time in this question. Here were given the submission off the minus one bell and ongoing Let you and plus one patrol royal and goes from zero up to infinity. It goes to the US I won and we continue. But in Finnish some on that we do. You find the pressure some s and consider submission. And it was from one Joe up to and only on this one girl and virtue and does one betrayal. And by formula here we know that the sm minus the 21 will be smaller equal than absolutely I am this one. We want this around here will be smaller than 7001 and we try that i to absolutely ico gender one off. Uh, you will be five factorial every computers and we have one divided by five factorial. Because you 1.4 times 10 power minus three. And this is greater than sopon 001 So we can try. And 83 because you want over the three and then we have to go to the seventh attire. And doesn't it go to no 1.9? Eight times £10 on this far, Anderson smaller than 0101 So from here, we can conclude that the ask three minus data. Uh, it would be the asked. Ju minus asked. Here it would be smaller. Then there's the Bonzo. Throw one, and then we can see that the submission of the minus one bell and what you m plus one factorial from zero to infinity. Approximately equal agenda. Uh, from zero to the Tu minus one NBA and over two and plus one Victoria and doesn't experience on again. The first term will be nah Guan on. Then the next time will be minus No one. Do you run by one? We have go to the three factorial and then we blast the one divided by the five TROIA Every computers. When we get the answer, he could you one mine us No. One, Almost six and then plus one over on the 720 So it will be equal Agenda Ah, 0.8 347 to to one of the 21 here it will be signed. One you go to the 0.84147 So the every ghost and this could be the answer.

Asked. Now we know by two that asked. Magnus asked him well, this morning than the next time You'll be envious. One. And we won this management of the area with smaller than Zougam's 01 And we can try with the and nickels. Do nothing unless I have the I d equals you and the square Last 3 20 Bunny hat Gondo one. Pour on the diesel from 00011 Still credit terms opens another one We can try with antiquity 31 and it's not the one with equal Jew 31 square mystery. So when we got better still be zero bomb Zozo 10 farts The great heavens open 01 So I am under 30. Joe Nathan the Jew ico ju ah, did you square plastic tree circle Joseph Bonzo zero ni seven is a smaller than turbans of the one. Therefore, the conclusion will be as minus asked 31. But this morning and your bones are the ones that means that community used 31 terms here

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.

In this question. We re Kanda from the Internet in threes. The parts of some months, the if it's some absolutely will be smaller. Vico, June, The next time I am this one and now in discussion were given the submission on the minus Qanbar. And on Then the van invited U N factorial Angus from zero to infinity. And we know the truth. Some for history's equity. It, of course I'm the one. And now we did notice. Anyway, Infinite sum and the brushes some s. And they go to the submission in this from search and understand power. And over the two and Victoria and then by formula we say that the s n minus the even It's some we smaller they go to the absolute. I am just one. We want this error here smaller than 0.1 Now it was tryingto I ah three. It will be equal to the one off. Three will be six factorial. Then we get Geico to one defining by six factorial in culture. The 1.389 10 10 by minus three. This one will be bigger than soap. On 001 We tried a far. Do we get that one hour eight factorial And this one will be equal to one dependent by a dictatorial. You go to the 2.48 times 10 by minus five. And since Manhattan's open 01 the phone From here we see that the ask three minus the asked Absolute value. This will be smaller and as Bonzo one. And from here we see that the infinite sum will be approximately equal to the submission from and reforms 02 the three minus Gwanda and ah, would you and Factorial we're computers on again. You go to the first time we go to the one. Ah, and then we have. Then next time will be equal to the minus one. And we have one off ju factorial, minus one other far factorial. And then isn't pluses minus 1/6 foot royal? Every computers one, we get the answer. He got you one plus Well, not just one hour Jew plus 1/24. Then we have minus one. Do you find him? By the 6 ft Royal On day we get Nico June. There's upon five Far Sarah 28 Where is true? one goes under one. Get in. Coachella's upon five. Father 30 to 3. So the M V closed? Yeah. Mhm. This will be the answer.


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