The Bessel function of order 0 is (a) Show that the series converges for all (b) Show
Chapter 9, Problem 71(choose chapter or problem)
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.
Text Transcription:
J_0 (x) = sum_k = 0 ^infty (-1)^k x^2k / 2^2k (k!)^2
x^2 J_0 ^prime prime + x J_0 ^prime + x^2 J_0 = 0
J_0
int_0 ^1 J_0 dx
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