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Determine whether the sequence converges or diverges. If it converges, find the limit.

$ a_n = \frac {n^2}{\sqrt {n^3 + 4n}} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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University of Michigan - Ann Arbor

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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.

01:27

Determine whether the sequ…

01:00

01:38

07:35

06:15

okay. We need to look at the Limited and goes to infinity of a m. Figure out whether or not that limit exists and whether or not it's fine. So this is limit as n goes to infinity of in squared over square root of in cubed plus foreign. This problem we do a similar trick to what we've done before As far as we look in the denominator, we looked to see what term in the denominator is going to infinity the fastest except here instead of dividing top and bottom by that term will need to factor out something. So we're gonna need to factor out this in cubed that we have happening here. So factor that out in the denominator and the square root function is multiplication. So we just gotta pull that outside like that. And then we have one plus for over and squared here. So now we can just rewrite this a little bit. Mhm in squared square root of n cubed is in to the 3/2. Mhm. Mhm. Okay, and now we can simplify this part a bit. So this is gonna be in to the Tu minus 3/2 so into the one half. And then we're still dividing by square root of one plus four over n squared. Okay, so hopefully that's legible. Okay, so now if we do that limit on top, divided by limit on bottom, which is something that we're allowed to do as long as we don't get indeterminate form, then we'll see that we're going to get infinity divided by one, which is just gonna be infinity. Yeah, right. So we're allowed to limit on top over limit on bottom as long as we don't get indeterminate form. Indeterminate form would be like infinity over infinity or something divided by zero. If we try doing that here, we're just gonna get infinity over one which is not considered indeterminate form. So that's fine. Okay, So the limit turns out to be infinity, so the limit exists, but it's not finite. So the sequence is still said to diverge

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