{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ma8_midterm_review_notes_2010

# ma8_midterm_review_notes_2010 - MATH 8 SECTION 1 MIDTERM...

This preview shows pages 1–3. Sign up to view the full content.

MATH 8, SECTION 1 - MIDTERM REVIEW NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Friday, Oct. 25th’s 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 we’ve 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 there’s not a lot to say here. However, it is worth it to mention the structure of an inductive proof again, as it’s been a while since we’ve 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 we’ve studied about sequences in this class is that of convergence: Definition 2.1. A sequence { a n } n =1 converges to some value λ if, for any distance , the a n ’s are eventually within of λ . To put it more formally, lim n →∞ a n = λ 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 λ . In symbols: lim n →∞ a n = λ iff ( )( N )( n > N ) | a n - λ | < . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ). When using these properties, please remember to show that both of the limits lim n →∞ a n , lim n →∞ b n exist before splitting them apart! TAs will dock you mad points for failing to do this, as it is one of the most common ways for people to make errors in limit calculations.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

ma8_midterm_review_notes_2010 - MATH 8 SECTION 1 MIDTERM...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online