Math 290 - Introduction to Analysis questions with detailed accurate answers and graded A+

Definition of a sequence - ANSA function from N to R epsilon definition of a limit - ANSFor all ε in R0, there exists N element of R such that for all n element of natural numbers, n N = |an - L| ε What does "the limit of an equals L" mean? - ANS"The limit of an equals L" means that when n is large, an is near L. The definition weaves together the largeness and nearness so as to obtain a precise meaning. You tell me ______ . I'll tell you ______. You'll find that _____. - ANSYou tell me how near you need it (epsilon). I'll tell you how far out in the sequence to look (N). You'll find that from that point on (n N), all of the terms an are as close to L as you asked for. This is what we mean by, the sequence eventually meets all error threshholds What does a limit proof from scratch tell you? - ANSA limit proof from scratch tells, for each ε, how big N must be, and proves that every n bigger than N really does satisfy |an − L| ε. What does a limit-to-limit proof verify? - ANSA limit-to-limit proof verifies a new limit by obtaining N from known limits, and proving that every n bigger than N does satisfy |an − L| ε. What does it mean for a sequence to diverge? - ANSFor all L element of R, there exists some ε (element of R 0) such that for all N element of R, there exists some n element of natural numbers where n N and | an - L | = ε Prove that if lim n→∞ an = L and lim n→∞ bn = M then lim n→∞ an + bn = L + M. - ANSLook at notes Be able to explain both the "maximum of two N's" and the "epsilon over 2" in terms of a subcontractor Prove that lim n-- infinity 1/n = 0 - ANSLook at notes Prove that if lim n-- infinity an = 1 and lim n-- bn = 2 then lim n-- infinity an + bn = 3 - ANSLook at notes What is the meaning of a series? - ANSSum(n =1 -∞) an