This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1.5 A vocabulary for sequences 1.5 27 A vocabulary for sequences Deﬁnition 1.5.1. A sequence of numbers is an countable ordered list a1 , a2 , . . . . Also, a
function a : N → R, where a(n) = an .
A sequence can be speciﬁed by giving
(i) the ﬁrst few terms: {1, 1 , 1 , . . . }
23
1
(ii) an explicit formula for the nth term: { n }, or (iii) a recurrence relation for the nth term: a1 = 1, an+1 = n−1
n an . Example 1.5.1. The Fibonacci numbers can be described by
(i) {1, 1, 2, 3, 5, 8, 13, 21, . . . }
(ii) 1
√
5 √
1+ 5
2 n − 1
√
5 √
1− 5
2 −n , or (iii) a0 = 1, a1 = 1, an+2 = an+1 + an .
Deﬁnition 1.5.2. {an } is increasing iﬀ an ≤ an+1 , ∀n.
{an } is strictly increasing iﬀ an < an+1 , ∀n.
{an } is increasing, (strictly increasing) iﬀ an ≥ an+1 (an > an+1 ), ∀n.
Deﬁnition 1.5.3. {an } is monotone iﬀ it is increasing or decreasing.
Deﬁnition 1.5.4. A sequence {an } is bounded above if there is a number B ∈ R such
that an ≤ B, ∀n. This B is an upper bound for the sequence {an }.
Deﬁnition 1.5.5. {an } is bounded below if there is a number B ∈ R such that an ≥ B, ∀n.
This B is an lower bound for the sequence {an }.
Deﬁnition 1.5.6. {an } is bounded iﬀ it is bounded above and bounded below.
Deﬁnition 1.5.7. {an } positive (negative), written an ≥ 0 (an ≤ 0) iﬀ {an } is bounded
below (above) by 0. 28 Math 413 Preliminaries ...
View
Full
Document
This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.
 Fall '11
 Wong

Click to edit the document details