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Sequences 2 CHAPTER 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 a1 is called the first term, a2 is the second term, and in general an is 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. Notation: The sequence { a1 ,a2 2 ,a3 ,…} is also CHAPTER denoted by {an} or {an}n=1 . 2.4 Continuity Definition: A sequence {an} has the limit L and we write: limn an = 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 limn an exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent). Theorem: If CHAPTER limn f(x) = L and2f(n)= an when n is an integer, then limn an = L. 2.4 Continuity Definition: A sequence {an} has the limit L and we write: limn an = 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 limn an exists, 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 2.4 Continuity c is a constant, then 1. limn (an + bn) = limn an + limn bn 2. limn (an - bn) = limn an - limn bn 3. limn can = c limn an 4. limn (an bn) = limn an . limn bn 5. limn (an / bn) = ( limn an / limn bn ) 6. limn c = c If an bn CHAPTER cn for n n0 and 2 limn an = limn cn = L, then limn bn = 2.4 Continuity L. Theorem: If limn |an|= 0, then limn an = 0. The sequence{rn}is convergent if –1 < r 1and divergent for all other values of r. limn rn = { 0 if –1 < r < 1 1 if r = 1 Definition: ACHAPTER sequence {an}is called 2 increasing if an < an+1 for all n 1, that is a1 < a2 < a3 < 2.4 Continuity …. It is called decreasing if an > an+1 for all n 1. A sequence monotonic if it’s either increasing or decreasing. Definition: ACHAPTER sequence {an}is bounded above if 2 there is a number M such that 2.4 Continuity 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. Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent.