Hw0sol - Math Brushup ST520 Hannig Adam Labadorf ST520 Hannig Math Brushup 1 ∞ X i =1 ∞ X j = i p j i The result of this equation depends on

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Unformatted text preview: Math Brushup ST520 - Hannig Adam Labadorf ST520 - Hannig August 27, 2007 Math Brushup 1. ∞ X i =1 ∞ X j = i p j + i The result of this equation depends on the value of p . If p < , p 6 =- 1 or p > 1 the equation will always diverge. If p =- 1 the value is unknown as the equation will oscillate between 0 and -1 infinitely. If ≤ p ≤ 1 the value can be found in closed form as follows: ∞ X i =1 ∞ X j = i p i p j ∞ X i =1 p i ∞ X j = i p j (1) Since ≤ p ≤ 1 , we can apply the closed form in Eq. (2): ∞ X k = m ar k = ar m 1- r (2) Our equation becomes: ∞ X i =1 p i p i 1- p 1 1- p ∞ X i =1 p i p i 1 1- p ∞ X i =1 ( p 2 ) i Again applying (2): 1 1- p p 2 1- p 2 (3) 2. Z 2 1 1 x dx = Z 2 1 x- 1 dx ln ( | x | ) 2 x =1 = ln ( | x | ) | x =2- ln ( | x | ) | x =1 = 0 . 6931 1 Math Brushup ST520 - Hannig 3. Z 1- 1 1 p | x | Z- 1 1 p | x | + Z 1 1 p | x | = 2 Z 1 1 p | x | 2 Z 1 ( | x | . 5 )- 1 = 2 Z 1 | x |- . 5 2 | x | . 5...
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This note was uploaded on 06/16/2009 for the course STAT 520 taught by Professor Hannig,j. during the Spring '09 term at Colorado State.

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Hw0sol - Math Brushup ST520 Hannig Adam Labadorf ST520 Hannig Math Brushup 1 ∞ X i =1 ∞ X j = i p j i The result of this equation depends on

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