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midtermI_21c_topics4

# midtermI_21c_topics4 - Calculus Math 21C Fall 2010 Summary...

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Calculus Math 21C, Fall 2010 Summary of basic results about sequences and series. 1. Sequences Definition of limit. A sequence { a n } converges to a limit L as n → ∞ , written lim n →∞ a n = L, or a n L as n → ∞ if for every ϵ > 0 there exists a number N such that | a n - L | < ϵ for all n > N. A sequence that does not converge is said to diverge. Diverges to infinity. A sequence { a n } diverges to , written lim n →∞ a n = , if for every number M there exists N such that a n > M for all n > N Note that such a sequence does not have a finite limit L , so it is divergent not convergent. Upper bound criterion. If { a n } is an increasing sequence ( a n +1 a n ) that is bounded from above (there exists a number M such that a n M for every n ) then { a n } converges. Sandwich theorem. If a n b n c n and the limits lim n →∞ a n = L, lim n →∞ c n = L exist and are equal, then lim n →∞ b n = L. Continuous functions of sequences. If lim n →∞ a n = L and f is a function that is defined on some open interval containing L and is continuous at L , then lim n →∞ f ( a n ) = f ( L ) .

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midtermI_21c_topics4 - Calculus Math 21C Fall 2010 Summary...

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