11.1 and 11.2 notes_Thomas' Calc II

# 11.1 and 11.2 notes_Thomas' Calc II - 11.1 Sequences...

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Unformatted text preview: 11.1 Sequences Infinite Seq. def.--an infinite seq. of #s is a function whose domain is the set of + integers Converges, Diverges, Limit def.--the seq. {an} converges to # L if to every + number ε there corresponds an integer N such that for all n , n > N => |a n − L | < ε If no such # L exists, then seq. {an} diverges. If {an} converges to L , we write lim n→∞ a n = L and call L the limit of the seq. Diverges to ∞ def.--the seq. {a n } diverges to ∞ if for every # M there’s an integer N such that for all n larger than N , a n > M . If condition holds we write Lim n→∞ a n = ∞ or a n →∞ Similarly if for every # m there’s an integer N such that for all n > N we have a n < m , then say {a n } diverges to -∞ we write Lim n→∞ a n = -∞ or a n → -∞ Theorem 1. Let {a n } and {b n } be seqs. of real #s and let A and B be real #s. If lim n→∞ a n = A and lim n→∞ b n = B then 1. Sum Rule: lim n→∞ (a n + b n ) = A + B. 2. Difference Rule: lim n→∞ (a n − b n ) = A − B. 3. Product Rule: lim n→∞ (a n∙ b n ) = AB. 4. Constant Multiple Rule: lim n→∞ (kb n ) = kB....
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11.1 and 11.2 notes_Thomas' Calc II - 11.1 Sequences...

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