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lec3 - EE 608 Computational Models and Methods Lecture 3...

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EE 608: Computational Models and Methods Lecture 3: Summations Read Appendix A of Introduction to Algorithms
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Summation and Product Formulas Given a sequence of numbers a 1 , a 2 , . . . , a n , the finite sum of those numbers can be written as n X 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 X k =1 a k , with the interpretation lim n →∞ n X 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 X k =1 | a k | converges in addition to X k =1 a k . The finite product of a 1 , a 2 , . . . , a n is written as n Y k =1 a k (if n = 0, the product is defined to be 1).
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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 X i =1 i As a notation for: polynomials: n X i =0 a i x i series: e x = X k =0 x k k !
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Summation Formulas and Properties Linearity: For any real number c and finite sequences, a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n : n X k =1 ( c * a k + b k ) = c n X k =1 a k + n X k =1 b k Linearity can also be used to manipulate summations with asymptotics (e.g., n X k =1 Θ( f ( k )) = Θ( n X k =1 f ( k ))). Manipulating Indices: Sometimes it useful to manipulate the limits of the summation. For example, n X k =1 a k +1 = n +1 X 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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Summation Formulas and Properties continued Arithmetic Series: Constant differences a k - a k - 1 .
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