This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 2 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University January 17, 2007 Jeff Linderoth IE170:Lecture 2 Sums Arithmetic Series 1 + 2 + + n = n k =1 k = n ( n + 1) 2 Sum Of Squares n k =0 k 2 = n ( n + 1)(2 n + 1) 6 Often, such formulae can be proved via mathematical induction Jeff Linderoth IE170:Lecture 2 Induction A way to prove that every statement in a (countably) infinite sequence of statements is true. How to do Induction 1 Prove that the first statement in the infinite sequence of statements is true: The base case . 2 Prove that if any one statement in the infinite sequence of statements is true, then so is the next one: The induction . Jeff Linderoth IE170:Lecture 2 More Sums Geometric Series n k =0 x k = 1 x n +1 1 x If  x  < 1 , then the series converges to k =0 x k = 1 1 x . Harmonic Series H n = 1 + 1 2 + 1 3 + + 1 k = n k =1 1 k ln( n ) Jeff Linderoth IE170:Lecture 2 Bounding Sums By Integrals When f is a (monotonically) increasing function, then we can approximate the sum n k = m f ( k ) by the integrals: n m 1 f ( x ) dx n k = m f ( k ) n +1 m f ( x ) dx. and a decreasing function can be approximated by n +1 m f ( x ) dx n k = m f ( k ) n m 1 For example, the harmonic series ( n k =1 k 1 ). n +1 1 x 1 dx n k =1 k 1 n x 1 dx ln( n + 1) n k =1 k 1 ln( n ) + 1 Jeff Linderoth IE170:Lecture 2 The Joy of Sets You are also responsible for knowing the definitions and notation of sets given in Appendix B : Empty Set Z : The set of integers: { 2 , 1 , , 1 , 2 } R : The set of real numbers R + : The set of nonnonnegative real numbers: { x R  x } A B x A x B A B x A such that x B  A  denotes the cardinality , or number of elements, of the set...
View Full
Document
 Spring '07
 Ralphs
 Systems Engineering

Click to edit the document details