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

The function $ J_1 $ defined by $ J_1(x) = \sum_{n = 0}^{\infty} \frac {(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} $is called the Bessel function of order 1. (a) Find its domain.(b) Graph the first several partial sums on a common screen.(c) If your CAS has built-in Bessel Functions, graph $ J_1 $ on the same screen as the partial sums in part (b) and observe how the partial sums approximate $ J_1. $

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

Campbell University

Idaho State University

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.

04:42

The function $J_{1}$ defin…

we use the ratio test here, so limit as n goes to infinity of absolute value of a in plus one over a N for this whole thing here is our A M value and we want for this to be less than one. This is limit as n goes to infinity of absolute value of X to the two times in plus one plus one over in plus one factorial times 10 plus two factorial times two to the two n plus one plus one and then notice that we didn't do anything with this minus one to the end. And that's because of the absolute value signs absolute value. It doesn't matter whether or not and it is positive or negative hopes. And this should be it should be a plus one there. Okay, so this is a n plus one, except for the minus one to the end, isn't there? And now we're dividing by a n So we're multiplying by the reciprocal of a M. So we'll have an infect. Oh, real top. We'll have an n plus one factorial up top as well, and we'll have a two times two to the end plus one up top. Okay, so whoops. And then we also have this next to the two n plus one down here. All right, so if we expand this, this is too in plus two plus one. So two n plus one will cancel out with this two n plus one. So these powers of X are just going to simplify the X squared in factorial divided by N plus one factorial is going to simplify the one over n plus one in plus one factorial divided by in plus two factorial is one over n plus two, two n plus one plus one. This is the same thing as to one plus two plus one, 21 plus one is going to cancel it with this two n plus one. So then we're just going to have to to the to and the denominator. So it doesn't matter what X is. As n goes to infinity, this is going to go to zero, which is less than one. So we get convergence regardless of what X is. So the domain is everything minus infinity to infinity. So if we look at the somewhere we have, let's see s one of X So this is this guy when we plug in an equal zero plus this guy, when we plug in n equals one and if you graph it, you know, you get something kind of squiggly. This should be passing through. The origin is something kind of squiggly like this Here, this is somewhere around three. This is somewhere around 0.6. Okay, so that's, you know, that'd be the second partial sum you took. You could also have the partial somewhere. You're just looking at the value you get when it is equal to zero and, uh, the whole function from in equal 02 Infinity is gonna look kind of like this, except it's going to be more squiggly. Okay, so it's still going to have this type of thing happening. Except now we have more squiggles. So in both directions and again, this value is going to be some were pretty close to three and appear this is somewhere pretty close to 0.6. So you'll see that the partial sums do start to look like the function that we're evaluating. And the more partial sums that you do, the more squiggly it's going to get, and then eventually it's going to turn out to be this really squiggly thing here

View More Answers From This Book

Find Another Textbook

01:01

Which of the following about the binomial distribution is NOT atrue stat…

04:51

The times per week a student uses a lab computer are normallydistributed…

01:02

Given the class limits below, what is the class width?1-5, 6-10, 11-15, …

02:14

1. The general form of a sine or cosine graph appears asfollows:f (x…

04:50

The city of Tucson, Arizona, employs people to assess the valueof homes …

A charter flight charges a fare of $300 per person plus$4545per person f…

01:49

Find the standard deviation for the given sample data. Round your answer to …

05:41

Solve the differential eqaution with a method of substitution ofhomogene…

03:22

{Exercise 4.45(Algorithmic)}In an article aboutinvestment al…

04:32

Let X be normally distributed withmean μ = 4.0 and standarddeviation…