Math 117 Midterm Review Notes

# Math 117 Midterm Review Notes - 1.1 Sequences and...

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1.1 Sequences and Convergence Sequence : A sequence is a function whose domain is the set of positive integers, the natural numbers. A sequence converges to a real number A, if and only if, for each ε > 0, there is a positive integer, N, such that for all n < N ( little n greater than or equal to big N) , we have the inequality |a n – A| < ε A neighborhood is a set of real numbers , Q, such that Q contains an interval of positive length centered at a real number A, that is if and only if, there exists ε > 0 such that (A - ε , A + ε ) is in Q. Lemma: A sequence converges to A if and only if each neighborhood of A contains all but a finite number of terms from the sequence a.k.a. each neighborhood must contain infinitely many terms. Theorem 1.1 – If {a n } from n=1 to converges to A and also to B, then A = B. Theorem 1.2 – If {a n } converges to A, then {a n } is bounded. 1.2 Cauchy Sequences A sequence is considered Cauchy if and only if for each ε > 0 there is a positive interger

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Math 117 Midterm Review Notes - 1.1 Sequences and...

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