Problem 35 - Calculus: Early Transcendentals

Question

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 \( J_1. \)

Step-by-Step Solution

Answer

(a) Domain: all real numbers. (b) Plot partial sums for different \( m \). (c) Graph \( J_1 \) to show approximation by partial sums.

Step 1: Identifying the Domain

To find the domain of the Bessel function \( J_1(x) = \sum_{n = 0}^{\infty} \frac{(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \), we need to ensure the series converges for the values of \(x\). The expression \( x^{2n+1} \) suggests the series converges for all real \(x\) because it is an infinite polynomial series. Therefore, the domain of \( J_1(x) \) is all real numbers, \((-\infty, \infty)\).

Step 2: Plotting Partial Sums

A graph of the partial sums \( S_m(x) = \sum_{n = 0}^{m} \frac{(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \) for \( m = 1, 2, 3, \ldots \) provides insight into the convergence of the series. Use a computer algebra system (CAS) to plot these sums for different values of \(m\) over a reasonable range of \(x\), such as \([-10, 10]\). Note how the graphs change as \(m\) increases and approach the actual function \( J_1(x) \).

Step 3: Graphing the Actual Function

Use a CAS that includes built-in Bessel functions to graph \( J_1(x) \). Plot this on the same screen as the partial sums from Step 2. This comparison reveals how the partial sums approximate \( J_1(x) \). As \(m\) increases, the partial sums should align more closely with \( J_1(x) \) across the graph, demonstrating the convergence of the series to the Bessel function.

Key Concepts

Convergent Series

In mathematics, a convergent series is a series whose partial sums tend to a finite limit as they progress to infinity. This means the terms of the series become smaller and effectively "settle down" to a specific value. For the Bessel function of order 1, represented by the series:\[ J_1(x) = \sum_{n = 0}^{\infty} \frac{(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \]the series converges for all real numbers, thanks to the fact that it is composed of increasingly smaller terms involving factorials. The factorial term in the denominator grows rapidly compared to the polynomial terms in the numerator, which helps ensure that the entire expression approaches zero. This fundamental property leads us to determine the domain of the Bessel function as all real numbers \[ (-\infty, \infty). \]

Partial Sums

Partial sums are a vital concept when dealing with infinite series. They represent the sum of the first several terms of a series, providing an approximation of the series' total sum. For the Bessel function, partial sums are defined as:\[ S_m(x) = \sum_{n = 0}^{m} \frac{(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \]where \( m \) indicates the number of terms included in the sum. As \( m \) increases, these sums approach the actual value of the Bessel function. Partial sums are useful because they allow us to study the behavior of a function at various stages of convergence. When graphed, they show how closely the summed terms approximate the true function as more terms are included.

Graphing with CAS

Computer Algebra Systems (CAS) are powerful tools that assist in mathematical calculations and visualizations. By graphing the partial sums of a series like the Bessel function, we can analyze the convergence behavior of the function. Using CAS, you can visually compare the partial sums plotted for different values of \( m \) on domains such as \([-10, 10].\)To visualize the convergence process:

• Graph each partial sum for increasing values of \( m \).
• Observe how these plots change and gradually align more closely with the Bessel function.

This method allows us to clearly see how the function is constructed term-by-term and how it approximates the true Bessel function as more terms are added.

Domain of a Function

The domain of a function refers to all the possible input values (or x-values) for which the function is defined. For the Bessel function of order 1, the domain is determined by the convergence of the series that defines it. Given:\[ J_1(x) = \sum_{n = 0}^{\infty} \frac{(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \]The series converges for all real numbers. The absence of division by zero, negative square roots, or other common domain restrictions means that any real number can be used as an input. Hence, the domain of the Bessel function is \((-\infty, \infty).\)Understanding the domain is crucial for ensuring that the function is applicable in the context of various mathematical problems and applications.