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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?

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

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Missouri State University

Campbell University

Harvey Mudd College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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02:02

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

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