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Discrete Mathematics with Graph Theory (3rd Edition) 205

Discrete Mathematics with Graph Theory (3rd Edition) 205 -...

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Section 7.7 (b) nDn_l+(-I)n n[ (n -1)!(1- n + ~ - ... + (_I)n-l (n 2 1)!)] + (_I)n n!(I- n + ~ - ... + (-I)n~) - n!(-I)n~ + (_I)n Dn - (_I)n + (_I)n = Dn (c) Dl = 0 while the values of D 2,D3, D4, D5 appear in the text. Using (b) D6 = 6D5 + 1 = 6(44) + 1 = 265; D7 = 7D6 -1 = 7(265) -1 = 1854; Ds = 8D7 + 1 = 8(1854) + 1 = 14,833; Dg = 9Ds - 1 = 133,496; DlO = 10Dg + 1 = 1,334,961. The information requested is tabulated below. n Dn Dn/n! n Dn 1 0 0 6 265 2 1 .5 7 1854 3 2 .333333 8 14,833 4 9 .375000 9 133,496 5 44 .366667 10 1,334,961 Note that ~ ~ .367879. 2 3 12. By equation (4) of this section, e- x = 1 - ~ + ~! - ~! + ... and by (12) of Section 5.4, _1_ = 1 + x + x 2 + x 3 + ... I-x e- X Thus
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Unformatted text preview: , I-x = (1 _ .::. + x 2 _ x 3 + ... ) (1 + x + x 2 + x 3 + ... ) I! 2! 3! 1 2 1 1 3 1 + Ox + (1 - 1 + 2!)x + (1 - 1 + 2! -3!)x + ... +(1-1 + ~ -~ + ... + (_I)n~)xn + ... Dn/n! .368056 .367857 .367882 .367879 .367879 and we note that for any n ;:::: 0, the coefficient of xn is ~ by Proposition 7.6.2. Exercises 7.7 1. (a) [BB] (x + y)6 x 6 + 6x 5 y + 15x 4 y2 + 20x 3 y3 + 15x 2 y4 + 6xy5 + y6 (b) [BB] (2x + 3y)6 = (2x)6 + (~)(2x)5(3y) + (~)(2x)4(3y)2 + (~)(2x)3(3y)3 +(~)(2X)2(3y)4 + (~)(2x)(3y)5 + (3y)6 64x 6 + 6(32x5) (3y) + 15(16x4)(9y2) + 20(8x3) (27y3) +15(4x2)(81y4) + 6(2x)(243y5) + 729y6 64x 6 + 576x 5 y + 2160x4y2 + 4320x 3 y3 + 4860x2y4 +2916xy5 + 729y6 203...
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