This preview shows page 1. Sign up to view the full content.
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 n1< x <= n. We have x = n if and only if x1 < 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) <= Ug(n)
for all n satisfying n >= n0
26 Big Ω
We define f(n) = Ω(g(n)) if and only if there
e...
View
Full
Document
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
 math

Click to edit the document details