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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 p-Series 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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