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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {n^4}{n^3 - 2n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Baylor University

University of Michigan - Ann Arbor

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.

02:16

Determine whether the sequ…

00:35

00:42

05:35

05:16

to figure out whether or not the sequence converges or diverges. I need to look at the limit as n goes to infinity of AM and figure out if that's the women exists. And if it's a finite number so we can do the trick where we look at the biggest power of in and the denominator and divide the top in the bottom, I at number so we can divide the top on the bottom by in cubed, and we have been on top than one minus two over and squared in the denominator. And as long as you don't get something an indeterminate form, you're allowed toe put the limit on top and the limit on the bottom. So as long as you don't get something like infinity over infinity or something divided by zero, then your levity. This here. We're just going to get infinity over one, so that's fine. That's not indeterminant form. It's just infinity. The limit does exist, but infinity is not a real number, so the limit didn't go to anything finite. So we say that the sequence diverges

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