Infinite Sequences and Series

# Infinite Sequences and Series - Infinite Sequences and...

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Unformatted text preview: Infinite Sequences and Series Calculus 2 (Wheeler) - The University of Pittsburgh Fall 2008 MWF 9:00-9:50 (Course #10052) Room 426 Benedum MWF 2:00-2:50 (Course #12728) Room 525 Benedum 1. Definitions and Basics Definition 1.1 (Sequence) . A sequence { a n } is an ordered list of numbers. Theorem 1.2. Let { a n } be a sequence. If lim n →∞ f ( x ) = L and if f ( n ) = a n whenever n is a positive integer, then lim n →∞ a n = L . Note that by this theorem all of our limit laws for functions now apply to sequences. As well, we have a Squeeze Theorem for Sequences: Theorem 1.3 (Squeeze Theorem for Sequences) . If a n ≤ b n ≤ c n for n ≥ n where n is a positive integer and if lim n →∞ a n = L = lim n →∞ c n , then lim n →∞ b n = L . Definition 1.4. If a n < a n +1 for all n ≥ 1 , then { a n } is called an increasing sequence . As well, if a n > a n +1 for all n ≥ 1 , then { a n } is called an decreas- ing sequence . A sequence that is increasing or decreasing is called a monotonic sequence Definition 1.5. If there exists a real number M such that a n ≤ M for all n ≥ 1 , then { a n } is said to be bounded above . If there exists a real number N such that a n ≥ N for all n ≥ 1 , then { a n } is said to be bounded below . A sequence that is both bounded above and bounded below is said to be a bounded sequence . Theorem 1.6 (Monotonic Sequence Theorem) . Every bounded, monotonic sequence is convergent. Definition 1.7 (Series) ....
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Infinite Sequences and Series - Infinite Sequences and...

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