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

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

Oregon State University

Idaho State 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.

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Determine whether the sequ…

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for this problem will use the Squeezed Sam. So there thing that kind of gets tricky with this. A in term is this co sign squared of N to the coastline squared Event is the troublemaker here, and hopefully it's something that's bounded. And if he knows which is the case here, then you can use the squeeze there. So we bound this thing that's giving us some trouble. This's co sign squared of ends that's going to be bigger than or equal to zero and less than or equal to one. You can't be a negative number because this is co sign squared. Okay, so this is going to be something that's true for all in, and we can divide by two to the end. And it's not going to change these bounds either, because to the end is always going to be some positive. Number two, divided by a positive number, are multiplying by a positive number. We don't have to worry about flipping these things that are, and then we can put these limits on take the limit as n goes to infinity limit as n goes to infinity. No Lamma as n goes to Infinity case is on the far left. This is still zero. Now we have the limit in question here by construction. This was our way in term. And then on the right hand side, we have limited as n goes to infinity of one over to the end. So that is also going to go to zero. So this Lim is trapped between zero and zero for it squeezed between zero and zero. That's where the name the serum comes from. So the only logical conclusion is that limit, as in approaches, infinity of a N is actually equal to zero to our sequence does converge and it converges to zero.

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