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Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.$ \left\{\begin{array} 1, 0, -1, 0, 1, 0, -1, 0, . . . .\end{array}\right\} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Campbell University

Oregon 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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Find a formula for the gen…

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Find a formula for the $n$…

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for this problem. It looks like there's not anything super nice happening. Looks like the odd terms. We're all going to be zero and the even terms they're going to be switching off between minus one at one. So it might be hard. Teo, write all of that information and one compact way, But you can always do it as this just writing it piece wise. So Anne is going to be zero if in is odd and it will be minus one to the in, divided by two if n is even right because we know that they even terms we're going to be switching between minus one and one. So if we just had minus one to the end happening here since when is even if we're looking at this bottom part than minus one to the end would always be a positive number. So that wouldn't be quite what we wanted, Since it is even we should be allowed to divide by two like we are here. You, Khun, we can check within his one. Then that means in his odds, we should get zero. So that agrees with what we have here. If it is too then we would have something. Even so, we'd be using this bottom part. We have minus one of the two over to which is just minus one. So that's good in equals. Three works in equals for Let's just check that in equals four. We would again use the this bottom part minus one to the four over too, would give us ah minus one squared, which is positive one, which is what we want. So this equation looks like it holds up.

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