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**Unformatted text preview: **MATH 8, SECTION 1, WEEK 2 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Friday, Oct. 8th’s lecture. In this talk, we study sequences. 1. Random Question Question 1.1. First, prove that you cannot cover R with disjoint circles of positive radii. Then, find a way to cover R 3 with disjoint circles of positive radii! 2. Sequences: Working from the Basics In our last lecture, we introduced the notion 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 | < . Most people are generally pretty good with developing an “intuition” for what convergence means; when it comes to actually proving that a sequence converges, however, it’s easy to get confused. How do you find your N ? What does it mean to have actually proved convergence? In general, proofs that a given sequence { a n } ∞ n =1 converges to some value L will go as follows: • First, examine the quantity | a n- L | , and try to come up with a very simple upper bound that depends on n and goes to zero. Example bounds we’d love to run into: 1 /n, 1 /n 2 , 1 / log(log( n )) . • Using this upper bound, given > 0, determine a value of N such that whenever n > N , our simple bound is less than ....

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