Intro to Analysis in-class_Part_11

# Intro to Analysis - 1.5 A vocabulary for sequences 1.5 27 A vocabulary for sequences Deﬁnition 1.5.1 A sequence of numbers is an countable

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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 ...
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## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis - 1.5 A vocabulary for sequences 1.5 27 A vocabulary for sequences Deﬁnition 1.5.1 A sequence of numbers is an countable

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