Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

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

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

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

$$[-1,1)$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Missouri State University

Harvey Mudd College

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.

03:14

Find the radius of converg…

04:51

01:54

02:45

02:40

02:12

Okay. The radius of convergence are is just the limit, as in goes to infinity of the absolute value of a N over A and plus one where one over two and minus one is going to be our way in term. So this is Lim has n goes to infinity of two times in plus one minus one over two and minus one because dividing by something is the same thing as multiplying by the reciprocal. So that's how this two times in plus one minus one ended up being on top here. Okay, so this is something that's linear on top, something that's linear on the bottom. And the leading coefficient is too in both these cases. So this limit just turned out to be one. Okay, And now we just need to figure out for the interval of convergence, whether or not one in minus one get included. So when acts is equal to minus one, we need to figure out what happens next is minus one. We have minus one to the end over two and minus one can. This is from n equals one to infinity to buy the alternating signed test. This does give us convergence. It will include that. And then for X equals one, we have some from n equals one to infinity of one over two and minus one. And if we make the denominator something that's bigger, we should get something smaller. So this has to be bigger than one over to N. And this is half of the harmonic Siri's so half of infinity, which is still infinity. So this diverges case, we do not include one, but we do include minus one. So our interval of convergence is minus one and we are including that so close bracket and one give us divergence, so we're not including that.

View More Answers From This Book

Find Another Textbook

00:51

1.A circular mirror has a radius of 28cm. find its area.

01:30

LEARNING TASK 1:QUIZ: Direction: Choose the letter of the correct answer. Wr…

01:02

the side of a cube are numbered 11 to 16 if john lens rolled the cube once w…

01:03

which is the correct conversion of 36.371° to the nearest minute?

05:28

Learning Tak : Solve each problem. Draw an illustration if necessaryA desk d…

02:26

1. Susan saves money for her birthday cake. Which of the following arethe va…

01:00

a garden inside park has the shape of a trapezoid.its bases are 12 meters an…

03:38

1. Marilyn’s mother bought 5 reams of bond paper to be used for their printi…

00:39

Read and understand the problem,then answer what is being asked. A playgroun…

01:12

If the volume of the given prism is 186 m3 then what is the volume of the py…