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hw0sol - Math Brushup ST520 Hannig Adam Labadorf 1 ST520...

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Math Brushup ST520 - Hannig Adam Labadorf ST520 - Hannig August 27, 2007 Math Brushup 1. i =1 j = i p j + i The result of this equation depends on the value of p . If p < 0 , p = - 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 0 p 1 the value can be found in closed form as follows: i =1 j = i p i p j i =1 p i j = i p j (1) Since 0 p 1 , we can apply the closed form in Eq. (2): k = m ar k = ar m 1 - r (2) Our equation becomes: i =1 p i p i 1 - p 1 1 - p i =1 p i p i 1 1 - p i =1 ( p 2 ) i Again applying (2): 1 1 - p p 2 1 - p 2 (3) 2. 2 1 1 x dx = 2 1 x - 1 dx ln ( | x | ) 2 x =1 = ln ( | x | ) | x =2 - ln ( | x | ) | x =1 = 0 . 6931 1
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Math Brushup ST520 - Hannig 3. 1 - 1 1 | x | 0 - 1 1 | x | + 1 0 1 | x | = 2 1 0 1 | x | 2 1 0 ( | x | 0 . 5 ) - 1 = 2 1 0 | x | - 0 . 5 2 | x | 0 . 5 0 . 5 1 x =0 = 2 | x | 0 . 5 0 . 5 x =1 - 2 | x | 0 . 5 0 . 5 x =0 = 4 4. 2 0 xe x dx Substituting: u = x, du = dx, v = e x , dv = e x dx 2 0 udv Applying integral of parts: uv - 2 0 vdu xe x - 2 0 e x dx xe x - e x = e x ( x - 1) 2 x =0 = 8 . 3891 5.
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