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Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $$ \displaystyle \sum_{n = 1}^{\infty} (x + 2)^n $

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Interval of convergence is $x \in(-3,-1)$The sum of the series is $-\frac{x+2}{x+1}$

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J Hardin

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Oregon State University

University of Michigan - Ann Arbor

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.

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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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Were given a series and We have to find the values of X.. this series converges and to find the some of the series For each value of X. So the series is the sum from n equals 1 to infinity of X-plus 2 To the ends. Power Notice that if X is some number, This is the form of a geometric series in particular with initial term X plus two. We'll call this a and common ratio our which is X plus to Also. Now we know that a geometric series converges If and only if The absolute value of the common ratio is less than 1. Therefore this is true if and only if the absolute value of X plus two is less than one, We can solve this inequality for X. This means X plus two must lie between one and negative one. In other words Xmas lie between on the one hand meg three and on the other hand negative one. These are the values of X for which it converges interval notation. X must lie in left parentheses, negative three, negative one right parenthesis. Now, for each of these values of X, we want to find this some. Well, we simply use the formula for the sum of a geometric series. This is the initial term which was X plus two over one minus the common ratio, Which is also X-plus 2. So this gives us X plus two over negative x minus one, Which we can factor out a negative and write this as negative times X plus two over X plus one. And this is the sum of these series.

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