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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}{\sqrt[3]{n}} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 8
Power Series
Sequences
Series
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.
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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okay for this problem. To get the radius of convergence, we take the limit as and goes to infinity, would you? Absolute value of hey n over and plus one. These guys here are a in terms, so we're going to get limit as n goes to infinity of absolute value of Q brute of n plus one over q brute of n So this is limit as n goes to infinity of the absolute value of Q brute of n plus one over in lemon is n goes to infinity of implicit one over in his one. So this is Cuban of one. So this is just one. So for the interval of convergence, we need to figure out whether or not we include one and whether or not we include minus one. So figure out what happens when X is equal to minus one and figure out what happened to next sequel, the positive one in here. So if access minus one, we have minus one of the end times a minus one of the ends that just gives us one up top. So then we would just have one over Hugh Brute of N. This is like into the one third power. One third is less than one. Therefore, this diverges. Okay, so we're not going to include minus one here. When X is equal to one, we get some from n equals one to infinity of minus one to the end over Cuba. It of n And this is going to converge by the alternating signed test. So we will include one. We're not gonna include minus one. So are you. Interval of Convergence is going to look like minus one comma one and one. We include minus one. We don't include
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