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ma017-2010-hw4

# ma017-2010-hw4 - ∞ X n =0 f n(1 n = e e(7 Let d be a real...

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MA017 2010 HOMEWORK 4 Due 2010-10-26 Tue. The Putnam problems on this set are 2, 6, 7, 8. (1) Show that n X k =0 ( - 1) k ( n k ) ( k + 1) 2 = 1 n + 1 n +1 X k =1 1 k . (2) Prove that all terms of the sequence a 1 = a 2 = a 3 = 1 , a n +1 = 1 + a n - 1 a n a n - 2 are integers. (3) Evaluate (for | x | < 1) X n =0 x 2 n 1 - x 2 n +1 . (4) Evaluate Y n =2 n 3 - 1 n 3 + 1 . (5) For 0 < a < b , deﬁne the sequences ( a n ) , ( b n ) by a 0 = a, b 0 = b, a n +1 = p a n b n , b n +1 = a n + b n 2 . (a) Prove that a n < a n +1 , b n > b n +1 and a n < b n for all n . (b) Prove that b n +1 - a n +1 = ( b n - a n ) 2 8 b n +2 . (c) Prove that for some g we have lim a n = lim b n = g, a n = g + O (1 /n 2 ) , b n = g + O (1 /n 2 ) . (6) Let f 0 ( x ) = e x and f n +1 ( x ) = xf 0 n ( x ) for n Z 0 . Show that
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Unformatted text preview: ∞ X n =0 f n (1) n ! = e e . (7) Let d be a real number. For each integer m ≥ 0, deﬁne a sequence { a m ( j ) } , j = 0 , 1 , 2 ,... by the condition a m (0) = d/ 2 m , and a m ( j + 1) = ( a m ( j )) 2 + 2 a m ( j ) , j ≥ . Evaluate lim n →∞ a n ( n ). (8) Evaluate ∞ X m =1 ∞ X n =1 m 2 n 3 m ( n 3 m + m 3 n ) . 1...
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