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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...
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This note was uploaded on 08/06/2008 for the course IE 170 taught by Professor Ralphs during the Spring '07 term at Lehigh University .
 Spring '07
 Ralphs
 Systems Engineering

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