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lec3-2 - Given a sequence of numbers a1 a2 an the nite sum...

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EE 608: Computational Models and Methods Lecture 3: Summations Read Appendix A of Introduction to Algorithms Given a sequence of numbers a 1 ,a 2 ,...,a n , the finite sum of those numbers can be written as n summationdisplay k =1 a k (if n = 0, the summation is defined to be 0). The infinite sum of a sequence of numbers a 1 ,a 2 ,... is written as summationdisplay k =1 a k , with the interpretation lim n →∞ n summationdisplay k =1 a k . If the limit doesn’t exist, the series diverges; otherwise, it converges. The terms of a convergent series cannot always be added in any order, but the terms of an absolutely convergent series can be rearranged. A series is absolutely convergent when summationdisplay k =1 | a k | converges in addition to summationdisplay k =1 a k . The finite product of a 1 ,a 2 ,...,a n is written as n productdisplay k =1 a k (if n = 0, the product is defined to be 1).
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ECE 608, Fall 2007 [ 2 ] Use of Summations Summations can be used: To compute the complexity (i.e., the running time) of loop constructs (e.g., for or while loops). To determine closed forms of recurrences. For example: x 1 = 1; x i = x i 1 + i x n = n summationdisplay i =1 i As a notation for: polynomials: n summationdisplay i =0 a i x i series: e x = summationdisplay k =0 x k k ! Linearity: For any real number c and finite sequences, a 1 ,a 2 ,...,a n and b 1 ,b 2 ,...,b n : n summationdisplay k =1 ( c a k + b k ) = c n summationdisplay k =1 a k + n summationdisplay k =1 b k Linearity can also be used to manipulate summations with asymptotics (e.g., n summationdisplay k =1 Θ( f ( k )) = Θ( n summationdisplay k =1 f ( k ))). Manipulating Indices: Sometimes it useful to manipulate the limits of the summation. For example, n summationdisplay k =1 a k +1 = n +1 summationdisplay l =2 a l Let l = k + 1, so k = l 1. Because 1 k n , 1 ( l 1) n , so 2 l n + 1.
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ECE 608, Fall 2007 [ 4 ] Summation Formulas and Properties continued Arithmetic Series: Constant differences a k a k 1 .
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