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(a) What is an alternating series?(b) Under what conditions does an alternating series converge?(c) If these conditions are satisfied, what can you say about the remainder after $ n $ terms?

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a) An alternating series is a series whose terms alternate between the positive sign and the negative sign.b) An alternating series converges $\sum a_{n}$ under the following two conditions:(1) $\left|a_{n+1}\right| \leq\left|a_{n}\right|,$ and $(2) \lim _{n \rightarrow \infty}\left|a_{n}\right|=0$c) The remainder after $n$ terms should be no more than absolute value of the very first term right after the $n$ th term. That is, $\left|R_{n}\right| \leq\left|a_{n+1}\right|$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Campbell University

Oregon State University

Idaho State University

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.

04:16

(a) What is an alternating…

06:08

(a) Decide whether the fol…

from party, eh? And all. Turning Siri's is the theories you write this out is a song whose terms alternate between positive and negative. So that's what all straining Siri's is for Part B. First, we should note that on alternating Siri's could always be written in the form this form right here or usually. Sometimes people like to write in this form, depending on the Siri's. So here, what we need is for being to be digging his ear off. We need that the limit of being a zero, and we need that The pianist are decreasing for all in. So here are conditions. So let's call these A, B and C are those already being used like he's a different amuse numbers here. Here's one two in three. So this is the answer for party under what conditions will on alternating Siri's converge well and also ending Syria's will converge when one through three are satisfied. So that's our answer for part two, and after part three, let's go ahead and look at the remainder. So if these conditions are satisfied, so that's assuming that the alternating Siri's converges. So here we have our ulcer, any Siri's, but since it doesn't matter what the starting point is, and this entire sum is equal to some number US. Bye, Assumption. We're saying that it converges from party now the remainder, after adding in terms, is given by s minus the impartial some. And this is by a theorem in your textbook alternating Siri's estimation there, Um, this says that the remainder satisfies this, but we know that in the limit that limit beyond equal zero. So this implies that the limit of our end equal zero. So this tells us that in the limit that the remainder, after adding in terms, goes to zero, and that's our final answer.

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