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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \ln(2n^2 + 1) - \ln(n^2 + 1) $

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The sequence converges to ln 2

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Harvey Mudd College

Baylor University

University of Nottingham

Boston 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.

00:46

Determine whether the sequ…

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01:01

for this problem when we're looking at the limit is n goes to infinity of am We're definitely not allowed to just plug in and equals infinity because then we'd have infinity here and we'd be subtracting infinity. So infinity minus infinity is indeterminate form doesn't mean that this limit doesn't necessarily exist. It just means that you're not allowed to do that. So something else that you could do would be to use properties of logs to know that subtraction on the outside corresponds to division on the inside. So instead of subtracting these two logs, we can write as one log the log of two in square posts, one divided by in squared plus one and the natural log function is continuous. So we're allowed to pull the limit inside of the function. And then we do the trick where we look at the denominator and we look at the term that's going to infinity, the fastest, and then we divide the top in the bottom by whatever that happens to be. So in this case we divide the top in the bottom by n squared to get two plus one over n squared, divided by one plus one over N squared, as in goes to infinity. This and this are both going to go to zero. So we just get natural log of two over one, which is something as natural log of two. So we converge and we converge to the natural log too.

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