Question

Bessel Function The Bessel function of order 0 is \(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{2^{2 k}(k !)^{2}}\) (a) Show that the series converges for all \(x\). (b) Show that the series is a solution of the differential equation \(x^{2} J_{0}^{\prime \prime}+x J_{0}^{\prime}+x^{2} J_{0}=0 .\) (c) Use a graphing utility to graph the polynomial composed of the first four terms of \(J_{0}\) - (d) Approximate \(\int_{0}^{1} J_{0} d x\) accurate to two decimal places.

Short answer

The series converges for all x. It also satisfies the given differential equation. The graph of the first four terms of \(J_{0}\) forms an oscillating pattern around 0. Approximation of the stated integral results in a finite value which is accurate up to two decimal places. Refer to the step-by-step solution for the detailed calculations and explanations.

Step-by-step solution

Series Convergence

To show the convergence of the series, one might apply the Ratio Test that states that if \( \lim_{k \rightarrow \infty} |\frac{a_{k+1}}{a_{k}}| < 1\), the series converges. Here, \( a_{k} = \frac{(-1)^{k} x^{2 k}}{2^{2 k}(k !)^{2}}\). Calculating the term-to-term ratio and evaluating the limit as k goes to infinity, we will find that it's below 1 for any x, thus the series converges.

Differential Equation

It's required to show that \(J_{0}\) satisfies the differential equation \(x^{2} J_{0}^{\prime \prime}+x J_{0}^{\prime}+x^{2} J_{0}=0\). One needs to calculate \(J_{0}^{\prime}\) (the first derivative of \(J_{0}\)) and \(J_{0}^{\prime \prime}\) (the second derivative of \(J_{0}\)) using the power rule for differentiation and derivating term by term. Upon substituting the derived expressions into the differential equation, one can safely deduce that it holds true.

Graphing

One can use any graphing utility to visualize the polynomial composed of the first four terms of \(J_{0}\). Those terms are: \(1 - \frac{x^2}{4} + \frac{x^4}{64} - \frac{x^6}{2304}\). The Bessel function \(J_{0}\) oscillates around 0, having a peak at \(x=0\).

Approximation of the Integral

To approximate \(\int_{0}^{1} J_{0} d x\) to two decimal places, one needs to use numerical methods such as the Trapezoidal rule or Simpson's rule to evaluate the integral. For simplicity, we can approximate \(J_0\) with just a few first terms of the series, recognizing that the integral should reduce to a simple polynomial integral. Evaluating this integral will yield an approximate value rounded to two decimal places.