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Unformatted text preview: Solutions to Homework 9, Math 55 Section 6.4 2. The generating function is 1 + 4 x + 16 x 2 + 64 x 3 + 256 x 4 = 4 n =0 (4 x ) n = 1 (4 x ) 5 1 4 x = 1 1024 x 5 1 4 x . 8. (a) Since ( x 2 + 1) 3 = (1 + x 2 ) 3 = 3 i =0 ( 3 i ) x 2 i , we have a n = ( 3 n/ 2 ) if 0 n 6 and n is even; otherwise, a n = 0. (b) Similarly, (3 x 1) 3 = 3 n =0 ( 3 n ) (3 x ) n ( 1) 3 n ; thus, if 0 n 3, then a n = ( 1) 3 n 3 n ( 3 n ) , and otherwise a n = 0. (c) We have 1 1 2 x 2 = k =0 (2 x 2 ) k = k =0 2 k x 2 k . Thus, if n is even, then a n = 2 n/ 2 , while if n is odd, then a n = 0. (d) The given function is equal to x 2 (1 x ) 3 . By the binomial theorem, this is equal to x 2 k =0 ( 3 k ) ( 1) k x k = k =0 ( 3 k ) ( 1) k x k +2 . Thus, to get a term x n we set k = n 2; if n < 2, this gives a n = 0, while if n 2, this gives a n = ( 3 n 2 ) ( 1) n . By the result of example 8 on page 438, we can write this as a n = ( n n 2 ) = ( n 2 ) . (e) The given function is equal to x 1 + n =0 3 n x n . Thus, a = 3 1 = 0; a 1 = 3 1 + 1 = 4; and if n 2, then a n = 3 n . (f) We rewrite 1+ x 3 (1+ x ) 3 = 1 3 x +3 x 2 (1+ x ) 3 = 1 3 x (1+ x ) 2 . This is equal to 1 3 x k =0 ( 2 k ) x k = 1 k =0 3 ( 2 k ) x k +1 . Thus, we have a = 1, while for n 1, we set k = n 1 to get the x n term, so a n = 3 ( 2 n 1 ) . Again using example 8, we can rewrite this as a n = 3 n ( 1) n . (g) If we multiply the top and bottom by 1 x , then this is equal to x x 2 1 x 3 = ( x x 2 ) k =0 x 3 k = k =0 ( x 3 k +1 x 3 k +2 ). Thus, we get a n = , if n (mod 3); 1 , if n 1 (mod 3); 1 , if n 2 (mod 3) . Alternately, let = 1+ i 3 2 ; then we can factor 1 + x + x 2 = (1 x )(1 x ), where = 1 i 3 2 is the complex conjugate. We can now use partial fractions to write x 1 + x + x 2 = 1 i 3 1 1 x 1 1 x = 1 i 3 X n =0 ( n n ) x n . Thus, a n = 1 i 3 ( n n ). Since = e 2 i/ 3 and = e 2 i/ 3 , this can be rewritten in terms of real numbers as a n = 2 3 sin( 2 n 3 ). (h) The given function expands to 1 + k =0 (3 x 2 ) k /k ! = 1 + k =0 (3 k /k !) x 2 k . Thus, if n is odd, then a n = 0; a = 1 + 1 = 0; and if n 2 is even, then a n = 3 n/ 2 / ( n/ 2)!....
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 Spring '08
 STRAIN
 Math

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