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Sequences. CHAPTER 2. 2.4 Continuity. A sequence can be thought of as a list of numbers written in a definite order: a 1 ,a 2 ,a 3 ,a 4 …,a n ,… The number a 1 is called the first term, a 2 is the second term, and in general a n is the nth term.
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Sequences CHAPTER 2 2.4 Continuity A sequence can be thought of as a list of numbers written in a definite order: a1 ,a2 ,a3 ,a4 …,an ,… The number a1is called the first term, a2is the second term, and in general anis the nth term. For every positive integer n there is a corresponding number an and so a sequence can be defined as a function whose domain is the set of positive integers.
CHAPTER 2 Notation: The sequence { a1 ,a2 ,a3 ,…} is also denoted by {an} or {an}n=1 . 2.4 Continuity Definition: A sequence {an} has the limit L and we write: limnan = L or an L as n if we can make the terms an as close to L as we like by taking n sufficiently large. If limnanexists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).
CHAPTER 2 Theorem: If limnf(x) = L and f(n)= anwhen n is an integer, then limnan = L. 2.4 Continuity Definition: A sequence {an} has the limit L and we write: limnan = L or an L as n if we can make the terms an as close to L as we like by taking n sufficiently large. If limnanexists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).
CHAPTER 2 • If {an} and {bn} are convergent sequences and c is a constant, then • limn (an + bn) = limnan + limn bn • limn (an - bn) = limn an - limn bn • limncan = c limn an • limn (an bn) = limnan . limn bn • limn (an / bn) = ( limnan / limnbn ) • limnc = c 2.4 Continuity
CHAPTER 2 If an bn cnfor n n0 and limnan = limncn = L, then limnbn = L. 2.4 Continuity Theorem:If limn|an|= 0, then limnan = 0. The sequence{rn}is convergent if –1 < r 1and divergent for all other values of r. limnrn = { 0 if –1 < r <1 1 if r = 1
CHAPTER 2 Definition: A sequence {an}is called increasing if an < an+1for all n 1, that is a1 < a2 < a3 < …. It is called decreasing if an > an+1for all n 1. A sequence monotonic if it’s either increasing or decreasing. 2.4 Continuity
CHAPTER 2 Definition: A sequence {an}is bounded above if there is a number M such that an M for all n 1. It is bounded below if there is a number m such that m an for all n 1. If it is bounded above and below, then {an}is a bounded sequence. 2.4 Continuity Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent.