Find the domain of the Bessel function of order 0 defined by the following. Solution 8 Jo(x) = (-1)x2n (-1)x2n n = 0 2²n (n!)² Let an lim n→∞ = [22n (n!)²] Then we get the following. an+1 n (-1) + 1x2(n + 1) = lim 22(n+1)(n+1)!² n-→∞ = lim +27 +2 22n (n!)2 (-1)x2n 22n (n!)2 x2n = 0 n→ 22n + 2(n + 1)²(n!)² = lim n→∞ < 1 for all x Thus, by the Ratio Test, the given series converges for all values of x. In other words, the domain of the Bessel function Jo is (−∞, ∞) = R.
Find the domain of the Bessel function of order 0 defined by the following. Solution 8 Jo(x) = (-1)x2n (-1)x2n n = 0 2²n (n!)² Let an lim n→∞ = [22n (n!)²] Then we get the following. an+1 n (-1) + 1x2(n + 1) = lim 22(n+1)(n+1)!² n-→∞ = lim +27 +2 22n (n!)2 (-1)x2n 22n (n!)2 x2n = 0 n→ 22n + 2(n + 1)²(n!)² = lim n→∞ < 1 for all x Thus, by the Ratio Test, the given series converges for all values of x. In other words, the domain of the Bessel function Jo is (−∞, ∞) = R.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![Find the domain of the Bessel function of order 0 defined by the following.
Solution
8
Jo(x) = (-1)x2n
(-1)x2n
n = 0 2²n (n!)²
Let an
lim
n→∞
=
[22n (n!)²]
Then we get the following.
an+1
n
(-1) + 1x2(n + 1)
= lim 22(n+1)(n+1)!²
n-→∞
=
lim
+27 +2
22n (n!)2
(-1)x2n
22n (n!)2
x2n
=
0
n→ 22n + 2(n + 1)²(n!)²
=
lim
n→∞
< 1
for all x
Thus, by the Ratio Test, the given series converges for all values of x. In other words, the domain of the Bessel function Jo is (−∞, ∞) = R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74f6cd98-5612-491b-92c1-2e44f23d515f%2Fdf886fa1-7b5b-485b-bb20-1a205eafa2d0%2Fsc19sx_processed.png&w=3840&q=75)
Transcribed Image Text:Find the domain of the Bessel function of order 0 defined by the following.
Solution
8
Jo(x) = (-1)x2n
(-1)x2n
n = 0 2²n (n!)²
Let an
lim
n→∞
=
[22n (n!)²]
Then we get the following.
an+1
n
(-1) + 1x2(n + 1)
= lim 22(n+1)(n+1)!²
n-→∞
=
lim
+27 +2
22n (n!)2
(-1)x2n
22n (n!)2
x2n
=
0
n→ 22n + 2(n + 1)²(n!)²
=
lim
n→∞
< 1
for all x
Thus, by the Ratio Test, the given series converges for all values of x. In other words, the domain of the Bessel function Jo is (−∞, ∞) = R.
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