Discrete Mathematics with Graph Theory (3rd Edition) 139

Discrete Mathematics with Graph Theory (3rd Edition) 139 -...

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Section 5.2 137 38. (a) Let f(x) = x+x 2 +X 3 + . .. +xn. The desired sum is f'(x). Since f(x) is the sum of a geometric x(1 - xn) x - xn+l sequence with a = x and r = x, we have f(x) = 1 = 1 . Thus -x -x , [1 - (n + 1)xn](1 - x) - (x - xn+l)( -1) 1- (n + 1)xn + nx n + l f(x)= (1-x)2 = (1-X)2 . (b) As n -+ 00, Ixl n -+ 0, so the sum in (a) approaches (1 ~ x)2' (c) LetS = 1+3x+5x 2 +7x 3 + ... be the desired sum and let SI = 1+2x+3x 2 +4x 3 + ... be the sum found in (b). WehaveS-SI = x+2x 2 +3x 3 +4x 4 + .. = x(1+2x+3x 2 +4x 3 + . . ) = XSI' 1+x Thus S = SI + XSI = (1 + X)SI = (1 _ x)2' (d) The desired sum is 1 + 3x(1 + 2x + 3x 2 + 4x 3 + . .. ) = 1 + 3XSl, where SI is the sum found . (b) S th h' 1 3x 1 + x + x 2 In part . 0 e sum ere IS + (1 _ x)2 (1- x)2 . 39 W< h 2k - 1 2k 1 11Th th . .. th d·fti . eave ak = 4 k - l = 4 k - 1 - 4 k - 1 = 2 k - 2 - 4 k - l ' us e sum In question IS e 1 erence between the sums of two geometric sequences. The first has a = 2~1 = 2, r = ! and the second has 1 1
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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