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Find the radius of convergence and interval of convergence of the series.
$ \sum_{n = 1}^{\infty} ( - 1)^n nx^n $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 8
Power Series
Sequences
Series
Campbell University
Baylor University
University of Nottingham
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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Find the radius of converg…
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02:57
Okay, So our radius of convergence, remember, Will you limit as n goes to infinity Absolute value of an over and plus one. These guys are R A in terms So this is going to be limit as n goes to infinity. Ah, see, just in over in plus one that's one. So our radius of convergence is just one for the interval of convergence. We want to see what happens when we plug in one into R sum and we want to figure out what happens when you plug in minus one into R sum So access minus one, then our terms they are just going to be minus one times minus one times in. So what is going to be something from in equals? One to infinity of n. So this is definitely going to diverge. And similarly, when X is equal to one, we're just going to have some from n equals one to infinity of minus one to the end times in. And that's still going to diverge because their terms are not even going to zero. So are in points are goingto be thrown out. Those aren't any good. So our interval of convergence we have to leave open like this interval of convergences interval from minus one toe, one open
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