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Jessica Wellington
University of Missouri - Columbia

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Sequences - Intro

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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Video Transcript

Okay, The first what the next time we're going to talk about is going to be sequences. And in the next videos, we're gonna answer one of the main questions about sequences, which is, Does the sequence converge or diverge? So that's gonna be the main focus of the next lessons that we do because we're gonna learn some different techniques on how toe answer this converge or diverge. Question. Um, some of the other things we're gonna can't talk about. We're going to cover some limit some. What's some common limit? So well, do some videos on common limits and some common common limit set ups that you should be familiar with to save yourself some time. And then we're also going to talk about the hospitals. Rules sandwich. The're, um So some techniques for answering that converge or diverge questions the techniques, techniques. We're going to be the sandwich the're, um, which we will cover. And also we'll talk about low hospitals rule which you guys air, probably familiar with it commonly used to find limits, and it works the same way with sequences. But we'll review Blaha Patel's Ruelas well, and then some other things. We're gonna look about. Look at our Cem different types of sequences. So we're gonna look at a recursive Lee define sequence, how to find the limit of these recursive sequences as well. We're going to talk about monotone and that serum that goes along with what a monotone sequences. So then we're going to kind of look at types of sequences and those types we'll talk about if it's monitor and then also, if it's non decreasing, non increasingly. So this is kind of the set up for what the next couple of lecture videos air going to be on. This is going to set a foundation for the remainder almost the remainder of this course, so make sure that you have a strong grasp on this section, for sure as we move forward.

Jessica Wellington
University of Missouri - Columbia
Calculus 2 / BC

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Series

Series Tests

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Root and Ratio Tests

Alternating Series Test

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Sequences
Using Sandwich Theorem and L'Hopital's

22:42

Sequences
Recursive Sequences

11:09

Sequences
Monotonic Sequence Theorem

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