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Find the radius of convergence and interval of co…

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Problem 3 Easy Difficulty

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

$ \sum_{n = 1}^{\infty} ( - 1)^n nx^n $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Related Topics

Sequences

Series

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Lectures

Video Thumbnail

01:59

Series - Intro

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.

Video Thumbnail

02:28

Sequences - Intro

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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Watch More Solved Questions in Chapter 11

Problem 1
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Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
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Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42

Video Transcript

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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Top Calculus 2 / BC Educators
Anna Marie Vagnozzi

Campbell University

Caleb Elmore

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Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

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.

Video Thumbnail

02:28

Sequences - Intro

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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