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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac { (\ln n)^2}{n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Oregon State University

Baylor University

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.

02:38

Determine whether the sequ…

for this problem when we're looking at limit as n goes to infinity of a n. Wendell to use low Patel's rule here. If you want to just plug in and equals infinity, then we'd have infinity on top and Cindy in the denominator as well. So were, of course, not allowed to just plug in infinity because infinity over INF India's indeterminate form. But if you are in the situation where you would get infinity over and Cindy, then low Patel's rule is applicable. Hello, petals. Rule is you know, when you're in this infinity over infinity or zero over zero type of setup. And when you're in that type of situation, you're allowed to just do the derivative of the numerator and divide by the derivative of the denominator. So the derivative of the numerator is too time's natural log of n to the two minus one, which is one times the derivative of the inside function. So times one over in through than the chain rule to this guy, and then we're dividing by the derivative of in with respect to end. So we're just dividing by one, and now we can rewrite this as limit as n goes to infinity of two times natural log of n divided by n and again. We're in a situation where we can use low Patel's rule. So this is limit as n goes to infinity of two times one over end divided by one. And this limit goes to zero. So our sequence does converge and it converges to zero.

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