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Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {( - 1)^n x^n}{n^2} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Oregon State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

03:00

Find the radius of converg…

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04:40

case, Remember, our radius of convergence are a cz limit as n goes to infinity absolute value of a N over A and plus one where these guys are a in terms. So we get Lim as in goes to infinity the minus one to the end that'Ll just get thrown out when we put these absolute value signs on So we're going to have absolute value of in plus one squared, divided by in squared because dividing by something of the same thing as multiplying by the reciprocal. That's how the plus one squared ended up on top and this limit is one. Now, for the interval of convergence, we need to figure out whether or not one or minus one works for convergence when we plug it into here. We just checked both of those individually when x is minus one, we have minus one of the end times minus one to the end, so that would just give us a positive one. And this is something that converges because the exponents, too, is bigger than once. This that's convergent. So that works and then X equals one. You have minus one to the end over in squared this is certainly going to converge. We've in fact, just shown that it'll converge. Absolutely so certainly will converge. Sex equals one also works. So both X equals minus one and X equals one. Both work. So we include both of those when we construct our interval of convergence. So when we include the points, we have these close brackets here.

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