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

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diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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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.

01:35

Determine whether the sequ…

04:26

01:12

once again we want to look at limit as n goes to infinity of AM case we haven't in happening in the denominator, we all said a square root of an happening in the denominator. When will do the trick where we divide the top in the bottom by something we wanted by the top on the bottom by determine the denominator that's going to infinity the fastest. So in that case, we're going to be dividing the top and the bottom by ten. Then we'LL just have minus one of the n plus one of top and one plus square root and divided by end. So that's one over square root of n. So this is limit as n goes to infinity of minus one to the end class one divided by limit as n goes to infinity of one plus one over square devyn and again Well, I wouldn't limit on top over limit on bottom as long as we don't get, you know, infinity over infinity or something divided by zero. So it's not happening here. Something else bad does seem to be happening here. This bottom limit just goes toe one, and we're just going to be stuck with Lim as n goes to infinity of minus one to the end plus one. And this limit is going to be trouble because it's just gonna be switching between positive one and negative one, so the limit does not exist and therefore are sequences said to diverge.

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