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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 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
if r = 1
(n + 1)a if r = 1
The case r = 1 holds, since arj = a for each of the
n + 1 terms of the sum.
The case r = 1 holds, since
(r − 1) j =0 arj =
arj +1 −
= j =0
j =0 arj − j =1
arj j =0 = ar
and dividing by (r − 1) yields the claim.
22 Sum of First n Positive Integers
For all n ≥ 1, we have
k = n(n + 1)/2
k=1 We prove this by induction.
Basis step: For n = 1, we have
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 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
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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.
- Fall '11