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# sol1 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

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Unformatted text preview: MATHEMATICS 3161: Fall 2009 (2009.9 - 2009.12) Assignment # 1 (Due Date: Sept. 23) 1. Determine the radius of convergence of the given power series. a) ∑ ∞ n =1 (2 x + 1) n n 2 b) ∑ ∞ n =1 ( x- x ) n n Solution : a) lim n →∞ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (2 x + 1) n +1 ( n + 1) 2 (2 x + 1) n n 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = lim n →∞ | 2 x + 1 | n 2 ( n + 1) 2 = | 2 x + 1 | , when | 2 x + 1 | < 1 , i.e.,- 1 < x < 0. Therefore, the radius of convergence is R = 1 2 . b) lim n →∞ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ( x- x ) n +1 n + 1 ( x- x ) n n vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = | x- x | , then when | x- x | < 1, the series converge, that is, R = 1. 2. Rewrite the given expression as a sum whose generic term involves x n . a) x ∞ summationdisplay n =1 na n x n − 1 + ∞ summationdisplay n =0 a n x n b) (1- x 2 ) ∞ summationdisplay n =2 n ( n- 1) a n x n − 2 Solution : a) x ∞ summationdisplay n =1 na n x n − 1 + ∞ summationdisplay k =0 a k x k = ∞ summationdisplay n =1 na n x n + ∞ summationdisplay n =0 a n x n = ∞ summationdisplay n =0 na n x n + ∞ summationdisplay n =0 a n x n = ∞ summationdisplay n =0 ( n + 1) a n x n b) (1- x 2 ) ∞ summationdisplay n =2 n ( n- 1) a n x n − 2 = ∞ summationdisplay n =2 n ( n- 1) a n x n − 2- ∞ summationdisplay n =2 n ( n- 1) a n x n = ∞ summationdisplay n =0 ( n + 2)( n + 1) a n +2 x n- ∞ summationdisplay n =0 n ( n- 1) a n x n = ∞ summationdisplay n =0 [( n + 1)(...
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sol1 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

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