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Unformatted text preview: ENGRD 270 Summer 2007 PROBLEM SET 1 – Solutions 1. Since the function F ( a ) = n X i =1 ( x i a ) 2 is twice continuously differentiable, if a minimizes F ( a ), then F ( a ) must be 0. Now F ( a ) = d da " n X i =1 ( x i a ) 2 # = n X i =1 d da ( x i a ) 2 = 2 n X i =1 ( x i a ) . So F ( a ) = 0 ⇒ n X i =1 ( x i a ) = 0 ⇒ n X i =1 x i = na ⇒ a = 1 n n X i =1 x i = x. Also note that F 00 ( a ) = d da " 2 n X i =1 ( x i a ) # = d da (2 na ) = 2 n > . Hence x will minimize F ( a ). 2.1 For λ ∈ (0 , 1], the series n X i =1 i λ will not converge, as n X i =1 i λ ≥ n X i =1 1 i ≥ n X i =1 Z i +1 i 1 x dx as x ∈ ( i, i + 1) ⇒ 1 x < 1 i ⇒ Z i +1 i 1 x dx ≤ 1 i = Z n +1 1 1 x dx = log( n + 1) , 1 and consequently lim n →∞ n X i =1 i λ ≥ lim n →∞ log( n + 1) = ∞ . 2.2 For λ > 1, the series n X i =1 i λ will converge, as n X i =1 i λ = 1 + n X i =2 i λ ≤ 1 + n X i =2 Z i i 1 1 x λ dx as x ∈ ( i 1 , i ) ⇒ 1 x > 1 i ⇒ Z i i 1 1 x...
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This note was uploaded on 04/17/2008 for the course ENGRD 2700 taught by Professor Staff during the Summer '05 term at Cornell University (Engineering School).
 Summer '05
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