Download the App!

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

Question

Answered step-by-step

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

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

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Gabriel Rhodes

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Missouri State University

Campbell University

Oregon 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.

03:07

Find the radius of converg…

01:54

02:45

03:00

03:14

02:12

02:16

figure out the radius of convergence. We figure out where we get convergence. So to figure out where we get convergence, we could do the ratio test on this thing by A N. I mean, this whole chunk here, including the X values So two X minus one to the n plus one. If we divide that by two X minus one to the end, then we're just going to be left with two X minus one. And then this will be the other stuff that we have after we do the algebra. Okay, And remember, in factorial is one times two times three times, not that all the way up to the Times, and so in, plus one factorial is going to cancel out within factorial. And the only thing that's going to be left over is in plus one. Okay. And for convergence, we're doing the ratio test here. We want for this to be something less than one. And now, if we we could notice that whatever we happen to put in for X here right as long as two x minus one is something that's finite and non zero two x minus one is non zero and we multiply it by something that blows up to infinity. Then we're going to get something that's infinite, an absolute value. So the only possible way to make this equality Hold this. This inequality holders, If two X minus one is equal to zero, we have to have that to X minus. One is equal to zero. And that's the only way that we're going to get this inequality tto be accomplished. Okay, so that means that two X is equal to one, so access one half. Okay, so the interval of convergence is just just the point, including one half just the interval, including one half here. Okay, so the length of this interval is one half minus one half. So this is this interval has no length to it. So that radius of convergence is zero and the interval of convergence is just the interval, including only one half

View More Answers From This Book

Find Another Textbook

02:25

commuter plane provides transportation from an international airport to the …

01:17

The points D(8, -3), E(5,5), F(-3,2) , and G(0, 6) form quadrilateral DEFG. …