{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec3-2

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

This preview shows pages 1–4. Sign up to view the full content.

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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
ECE 608, Fall 2007 [ 4 ] Summation Formulas and Properties continued Arithmetic Series: Constant differences a k a k 1 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}