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Show that aj +2 aj ↑ 217 Since h> 2, therefore, the series is convergent.

Show that aj +2 aj ↑ 217 Since h> 2, therefore, the series is convergent.

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+
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Test for convergence. Show that aj +2 aj Since h2, therefore, the series is convergent. 2 j
Transcribed Image Text:Test for convergence. Show that aj +2 aj Since h2, therefore, the series is convergent. 2 j
8.3 ODE Eigenvalue problems Example 8.3.1 Legendre Equation Ly(x) = (1-x²)y"(x) + 2xy'(x) = y(x). (8.23) Defines an eigenvalue problem when V2 spherical polar coordinates where x = cos -1 ≤ x ≤ +1. As boundary conditions, the solutions should be nonsingular at x = ±1. is separated in with the range Using Frobenius method, assume solutions of the form. From the assumed solution, Ariken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7 ed.) Elsevier We get 00 y' = [aj (s +j)xs+j-1, j=0 Substituting it in Eq. (8.23) Ž¶ -(1-x²) y = -(1-x²) v = Σ ²₁x j=0 Σ j=0 ajxs+j ajx³+j aj(s+j)(s+j-1)x+/-2 -aj(s +j) (s + j - 1)x+j-2 + a₁x²+1 = 0 00 (8.24) j=0 + 2x *-2 + 2x² Ariken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7 ed.). Elsevier. aj(s+j)(s+j-1)x+/-2 + 2x aj(s+j)(s+j-1)xs+j-2 20+0+2 +2²2 946 aj(s+j)x³+/ T=0 aj(s+j)(s+j-1)x+/+ 2x aj(s +j)x+j-1-2ax²+1 = 0 jo aj (s+j)(s+j-1)x+j+ 2a,(s+j)-2x+1 :0 ajx = 0. =o -ao(s) (s - 1)xs-²-a₁(s + 1)(s)xs-¹-aj(s+j)(s +j-1)xs+j-2 • Σ + =0 a(s+j)x³+j-1
Transcribed Image Text:8.3 ODE Eigenvalue problems Example 8.3.1 Legendre Equation Ly(x) = (1-x²)y"(x) + 2xy'(x) = y(x). (8.23) Defines an eigenvalue problem when V2 spherical polar coordinates where x = cos -1 ≤ x ≤ +1. As boundary conditions, the solutions should be nonsingular at x = ±1. is separated in with the range Using Frobenius method, assume solutions of the form. From the assumed solution, Ariken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7 ed.) Elsevier We get 00 y' = [aj (s +j)xs+j-1, j=0 Substituting it in Eq. (8.23) Ž¶ -(1-x²) y = -(1-x²) v = Σ ²₁x j=0 Σ j=0 ajxs+j ajx³+j aj(s+j)(s+j-1)x+/-2 -aj(s +j) (s + j - 1)x+j-2 + a₁x²+1 = 0 00 (8.24) j=0 + 2x *-2 + 2x² Ariken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7 ed.). Elsevier. aj(s+j)(s+j-1)x+/-2 + 2x aj(s+j)(s+j-1)xs+j-2 20+0+2 +2²2 946 aj(s+j)x³+/ T=0 aj(s+j)(s+j-1)x+/+ 2x aj(s +j)x+j-1-2ax²+1 = 0 jo aj (s+j)(s+j-1)x+j+ 2a,(s+j)-2x+1 :0 ajx = 0. =o -ao(s) (s - 1)xs-²-a₁(s + 1)(s)xs-¹-aj(s+j)(s +j-1)xs+j-2 • Σ + =0 a(s+j)x³+j-1
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