lecture2

lecture2 - IE170: Algorithms in Systems Engineering:...

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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 non-nonnegative 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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lecture2 - IE170: Algorithms in Systems Engineering:...

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