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

$ \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n5^n} x^n $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Missouri State University

Campbell University

Harvey Mudd 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.

02:40

Find the radius of converg…

01:11

02:57

11:13

03:01

03:21

05:58

04:40

the radius of convergence are is the limit as n goes to infinity of absolute value of a n over a n plus one were the ends are these terms here, So a and is minus one of the n minus one divided by n times five to the end. Once we write this in, we get one over n times five to be in and then dividing by 8 p.m. plus one is the same thing as multiplying by the reciprocal so multiplying by the reciprocal would get times n plus one times five to the end plus one. And keep in mind that this minus one of the n minus one gets thrown out because we're looking at absolute value here. So five to the end, plus one divided by five to the end is just five and limit as n goes to infinity of n plus one over n is one. So this turns into one times five, which is just five. So that's our radius of convergence. For the interval of convergence, we need to ask whether or not we include X equals five and whether or not we include X equals minus five so an X is minus five. The terms that we're going to be summing up are going to be minus one to the N minus one over in times five to the end and then multiplied by minus five to the end. Okay. And those, there's going to be some nice cancellations happening here. So minus one to the N minus one times minus one to the end is just minus one five to the end. Divided by five to the end is one. So this is going to be minus one over n. So where summing this up, this is going to be the minus version of the harmonic series. So that's going to diverge. Sex equals minus five. We will not include in our interval of convergence. And then when X is equal to five, we plug in X equals five in here and we get something similar. Except now we have minus one to the N minus one, divided by and and this is going to converge by the alternating sign test. So minus five, we get divergence. Five, we get convergence. So the interval of convergence we throw out minus five. We include five

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