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JH
Numerade Educator

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Problem 40 Medium Difficulty

How many terms of the series $ \sum_{n = 2}^{\infty} 1/[n(\ln n)^2] $ would you need to add to find its sum to within 0.01?

Answer

$\left\lceil e^{100}\right\rceil$

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Video Transcript

we're dealing with this infinite sum. And we'd like to know how many terms in a partial some congee use so that they air or the difference between the partial some sn hoops. Take a step back here say Chez equals to the end A end where this is our land we want to use We want to find the value of end such that this is true Where s is the entire sum And it's true for you up to this era right here, Cyril point zero one So motivated by r A n well, define this continuous version f I replacing and with X in the formula for a N and here will take ex to be at least two due to this condition over here and now recall the formula in the section for the air when using in terms as we are here. The remainder are n given by this satisfies this inequality. And now since we have f of X here, we want this to be Weston point zero one. So now it's a matter of solving this and finding end. So let's go ahead and replace F with the formula that we have up here and this becomes it becomes an integral that we can actually solve by using the U substitution. So here I would just go ahead and take you to be if you need to, right out to you. Just use that. But in either case, this becomes negative one over Ellen X And now let me pick this up Over here this becomes one over natural lot of end and I'll just go ahead and solved that friend. So in this case, we want into the larger than eats of one hundred. This may not be uninjured. So in that case, to be safe, let's just go ahead and take and to be greater than or equal to and we can write this This is the ceiling ceiling function and what it does is it This is not a singer, it increases it until it reaches the next singer. So by taking this value of N, we can ensure that in this case, let me write that out here partial some. This will be approximately equal to the original sum esse, and the air will be less than zero point zero one