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**Unformatted text preview: **MATH 8, SECTION 1 - MIDTERM REVIEW NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Friday, Oct. 25ths midterm review. The following is kind of a condensed, Cliffs-Notes-style review of the course thus far; here, we list all of the major theorems and results weve covered thus far, and talk a little bit about when we would use these theorems. Basically, this is the last four and a half weeks in one handout; if you want to see *examples* of how these things are actually used, consult the online notes for those particular weeks! 1. Proof Methods (Relevant lectures: Monday, wk. 1 , Wednesday, wk. 1 , Friday, wk. 1 .) Basically, you are all as a class quite capable with proof methods; so theres not a lot to say here. However, it is worth it to mention the structure of an inductive proof again, as its been a while since weve used induction (as opposed to direct proofs or proofs by contradiction!), and a lot of people get tripped up on the structure of these things: 1.1. Proofs by Induction. Suppose that you have a claim P ( n ) a sentence like 2 n n , for example. How do we prove that this kind of thing holds by induction? Well: we generally follow the outline below: Lemma 1.1. P ( n ) holds, for all n k . Proof. Base case: we prove (by hand) that P ( k ) holds, for a few base cases. Inductive step: Assuming that P ( m ) holds for all k m < n , prove that P ( n ) holds. Conclusion: P ( n ) holds for all n k . 2. Sequences (Relevant lectures: Friday, wk. 2 , Monday, wk. 3 , ) 2.1. Definitions. A sequence is just an infinite collection of objects { a n } n =1 indexed by the natural numbers. The main property that weve studied about sequences in this class is that of convergence: Definition 2.1. A sequence { a n } n =1 converges to some value if, for any distance , the a n s are eventually within of . To put it more formally, lim n a n = iff for any distance , there is some cutoff point N such that for any n greater than this cutoff point, a n must be within of our limit . In symbols: lim n a n = iff ( )( N )( n > N ) | a n- | < . 1 2 TA: PADRAIC BARTLETT 2.2. Tools. We have the following tools for manipulating and studying sequences: (1) Arithmetic and Sequences : Additivity of sequences : if lim n a n , lim n b n both exist, then lim n a n + b n = (lim n a n ) + (lim n b n ). Multiplicativity of sequences : if lim n a n , lim n b n both exist, then lim n a n b n = (lim n a n ) (lim n b n ). Quotients of sequences : if lim n a n , lim n b n both exist, and b n 6 = 0 for all n , then lim n a n b n = (lim n a n ) / (lim n b n )....

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