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We have x n if and only if x n x1 18 example prove

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Unformatted text preview: igns to a real number x the largest integer <= x. The ceiling function : R -> Z assigns to a real number x the smallest integer >= x. 3.2 -3.2 = 3 and = -4 and 3.2 =4 -3.2 = -3 17 Basic Facts We have x = n if and only if n <= x < n+1. We have x =n if and only if n-1< x <= n. We have x = n if and only if x-1 < n <= x. We have x =n if and only if x<= n < x+1. 18 Example Prove or disprove: ￿ √ ￿ ￿x￿￿ = ￿ x￿ 19 Example 3 ￿ Let m = ￿ ￿x￿￿ ￿ Hence, m ≤ ￿x￿ < m + 1 Thus, m2 ≤ ￿x￿ < (m + 1)2 It follows that m2 ≤ x < (m + 1)2 √ Therefore, m ≤ x < m + 1 √ Thus, we can conclude that m = ￿ x￿ This proves our claim. 20 Series and Sums You need to know - sequence notations - important sequences - summation notation - geometric sum - sum of first n positive integers 21 Geometric Series Extremely useful! If a and r ￿= 0 are real numbers, then ￿ n+1 n ar −a ￿ if r ￿= 1 j r −1 ar = (n + 1)a if r = 1 j =0 Proof: The case r = 1 holds, since arj = a for each of the n + 1 terms of the sum. The case r ￿= 1 holds, since n n ￿ ￿ ￿n (r − 1) j =0 arj = arj +1 − arj = j =0 n+1 ￿ j =0 arj − j =1 n+1 n ￿ arj j =0 = ar −a and dividing by (r − 1) yields the claim. 22 Sum of First n Positive Integers For all n ≥ 1, we have n ￿ k = n(n + 1)/2 k=1 We prove this by induction. Basis step: For n = 1, we have 1 ￿ k = 1 = 1(1 + 1)/2. k=1 23 Extremely. useful! Infinite Geometric Series Let x be a real number such that |x| < 1. Then ∞ ￿ k=0 1 x= . 1−x k 24 Asymptotic Notations You need to know - big Oh notation - big Omega notation - big Theta notation - be able to prove that f = O(g), ... - be able to show that O(g)=O(h) 25 Big Oh Notation Let f,g: N -> R be functions from the natural numbers to the set of real numbers. We write f ∈ O(g) if and only if there exists some real number n0 and a positive real constant U such that |f(n)| <= U|g(n)| for all n satisfying n >= n0 26 Big Ω We define f(n) = Ω(g(n)) if and only if there e...
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This test prep was uploaded on 03/24/2014 for the course CSCE 222 taught by Professor Math during the Fall '11 term at Texas A&M.

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