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Unformatted text preview: Sequences and Series Review Two very important (distinct) mathematical objects are sequences and se ries . A sequence is an infinite list of numbers, whereas a series is an infinite sum . With a series n =1 a n , though, we can associate two sequences: 1) its sequence of partial sums ( S n ), where S n = n k =1 a k 2) its sequence of terms ( a n ). Our primary concern with both sequences and series is convergence . A se quence ( a n ) converges if lim n a n = L for some real number L . A series converges if lim n n k =1 a k = L for some real number L ; this is the same as saying the sequence of partial sums ( S n ) converges. If a sequence or series does not converge, we say it diverges. We have developed many tools for determining whether a given sequence or series converges. The ultimate goal of all these tests is applying them to Taylor series. We know that if the Taylor series of f at a converges for a given value of x , then it converges to f ( x ). This is why we only test for convergence rather than try to evaluate these series: once we have convergence, we know right away what it converges to. Sequence Series What it is list sum Written ( a n ) n =1 a n a 1 ,a 2 ,a 3 ,... a 1 + a 2 + a 3 + Converges if lim n a n = L lim n n k =1 a k = L lim n S n = L Tests Limit Laws, Substitution Law n th Term Test Squeeze Law Integral Test Limits of Functions and Sequences Comparison Test Bounded and Monotone Limit Comparison Test LHopitals Rule Alternating Series Test Ratio Test Root Test pSeries Geometric Series 1 1 Sequences Limit Laws, Substitution Law These laws tell us how to compute limits of sequences. If we can determine what a sequence converges to, then trivially it converges. Recall the Substitution Law says if 1) lim n a n = L 2) f is continuous at L , then lim n f ( a n ) = f ( L ). Squeeze Law The Squeeze Law is very intuitive: if a n b n c n and lim n a n = lim n c n = L , then lim n b n = L also. Often we apply this with sin and cos, bounding them between 1 and 1. Limits of Functions and Sequences This is the idea that if lim x f ( x ) = L , then lim n f ( n ) = L also. For example, lim x 2 + 1 /x = 2, so lim n 2 + 1 /n = 2. This rule essentially exists to justify using LHopitals Rule on sequences. Bounded and Monotone By a theorem, any sequence ( a n ) which is bounded (for some positive number M , and for all n , M a n M ) and monotone (for all large enough n , either the a n s are increasing or decreasing) converges....
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 Spring '09
 MAYFIELD
 Calculus, Sequences And Series

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