hw4 - k x k =< x x> 1 2 1 2(a Prove that for any x y...

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Homework 4 – Math CS 120, Winter 2009 Due on Thursday, February 5th, 2009 1. Consider the Fibonacci sequence a 1 = a 2 = 1, and a n +1 = a n + a n - 1 , n 2 . Prove that lim n →∞ a n +1 a n exists, and find the limit. 2. Let x 1 = c , and for n 2 define the sequence { x n } by x n = 7 8 x n - 1 + 1 8 . For what real values of c does the sequence { x n } converge? To what does it converge? 3. Evaluate the limit lim n →∞ ± n - n e ± 1 + 1 n ² n ² . 4. Give a Cauchy sequence { x n } ⊂ R , and a continuous function f : R R , prove that the sequence { f ( x n ) } is a Cauchy sequence. 5. Evaluate lim n →∞ ³ n n 2 + 1 2 + n n 2 + 2 2 + ··· + n n 2 + n 2 ´ . 6. Let 0 < x 1 < y 1 . For n 1, set x n +1 = x n y n , y n +1 = 1 2 ( x n + y n ) . Show that lim x n and lim y n exist and are equal. ( Hint: Check that 0 < x n < y n , y n +1 < y n , and x n < x n +1 ) . 7. Assume that < · > : R n R is an inner product, and define the norm
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Unformatted text preview: k x k = < x , x > 1 / 2 . 1 2 (a) Prove that for any x , y ∈ R n , < x , y > = 1 4 ( k x + y k 2- k x-y k 2 ) . (b) Prove that k · k p does not derive from an inner product, unless p = 2, that is to say, there does not exist an inner product in R n such that k x k p = < x , x > 1 / 2 ∀ x ∈ R n , unless p = 2. 8. Given a set A ⊂ R n , and x ∈ R n , define the distance from x to A by d ( x , A ) = inf ± k x-y k ² ² ² ² y ∈ A ³ . Prove that for a closed set A , x ∈ A if and only if d ( x , A ) = 0....
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hw4 - k x k =< x x> 1 2 1 2(a Prove that for any x y...

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