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JH
Numerade Educator

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Problem 36 Hard Difficulty

(a) Find the partial sum $ s_10 $ of the series $ \sum_{n = 1}^{\infty} 1/n^4. $ Estimate the error in using $ s_10 $ as an approximation to the sum of the series.
(b) Use (3) with $ n = 10 $ to give an improved estimate of the sum.
(c) Compare your estimate in part (b) with the exact value given in Exercise 35.
(d) Find a value of $ n $ so that $ s_n $ is within 0.00001 of the sum.

Answer

a) 0.000$\overline{3}$
b) $s \approx 1.082328$
c) 0.0000053
d) $n=33$ terms for the required accuracy

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Video Transcript

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.