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# hw8key - AMATH 351 Homework 8 Due Section5.1 3 9 13...

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Unformatted text preview: AMATH 351 Homework 8 Due August 19, 2009 Section5.1 3, 9, 13 Section5.2 4, 12 Section5.4 6 Section5.5 10 Section5.6 6 Section5.7 7 Section 5.1 3. Determine the radius of convergence. ∞ X n =0 x 2 n n ! For the series to be convergent, lim n →∞ x 2( n +1) ( n +1)! x 2 n n ! = lim n →∞ x 2 n + 1 = 0 As indicated above, the ratio goes to for all the nite values of x , so the radius of convergence is ∞ . 1 9. sin x = x- x 3 3! + x 5 5!- x 7 7! + ··· = ∞ X n =0 (- 1) n x 2 n +1 (2 n + 1)! lim n →∞ (- 1) n +1 x 2 n +3 (2 n + 3)! × (2 n + 1)! (- 1) n x 2 n +1 = lim n →∞- x 2 (2 n + 3) (2 n + 2) = 0 for all the nite values of x , the radius of convergence is ∞ . 13. ln x = ln (1 + x- 1) = ( x- 1)- ( x- 1) 2 2 + ( x- 1) 3 3- ( x- 1) 4 4 + ··· = ∞ X n =1 (- 1) n +1 ( x- 1) n n By ratio test lim n →∞ (- 1) n +2 ( x- 1) n +1 n + 1 × (- 1) n +1 n ( x- 1) n = lim n →∞- n n + 1 ( x- 1) = | x- 1 | < 1 So radius of convergence is 1 . Section 5.2 4. y 00 + k 2 x 2 y = 0 , x = 0 , k a constant Since x = 0 is a regular point of the di erential equation, we assume y = ∞ X n =0 a n x n y = ∞ X n =1 na n x n- 1 y 00 = ∞ X n =2 n ( n- 1) a n x n- 2 2 Substitue into the DE, we get ∞ X n =2 n ( n- 1) a n x n- 2 + ∞ X n =0 k 2 a n x n +2 = 0 Shift the index for the both of the series, we get...
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hw8key - AMATH 351 Homework 8 Due Section5.1 3 9 13...

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