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# 10sSeq - SEQUENCES A sequence cfw_an is an ordered list of...

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SEQUENCES A sequence { a n } n =1 is an ordered list of numbers. Thinks of as { a n } n =1 = { a 1 , a 2 , a 3 , a 4 , . . . } goes on forever . You can also think of a sequence as a function f : N R with f ( n ) = a n . Def 11.1.1: limit of a sequence (intuition). A sequence { a n } has the limit L and we write lim n →∞ a n = L or a n L as n → ∞ if we can make the terms a n as close to L as we like by taking n sufficiently large. If lim n →∞ a n exists, we say the sequence converges (or is convergent ). Otherwise we say the sequence diverges (or is divergent ). Def 11.1.2: limit of a sequence (precise). A sequence { a n } has the limit L and we write lim n →∞ a n = L or a n L as n → ∞ if for each ε > 0 there is a corresponding integer N such that if n > N then | a n - L | < ε . In shorthand ε > 0 N N such that [ n > N = ⇒ | a n - L | < ε ] or equivalently ε > 0 N N such that [ n > N = L - ε < a n < L + ε ] Fun Compare Def 11.1.2 (page 677, limit of a sequence) to Def 2.6.7 (page 138, limit of a function).

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