Math 409-502Harold P. [email protected] theoremsSums, products, and quotientsIf an→ L and bn→ M then an+ bn→ L + M;and an· bn→ L · M;and if in addition L 6= 0 then bn/an→ M/L.[In the third case, an6= 0 when n is large, so bn/anmakes sense for n large.]Squeeze theoremIf an≤ bn≤ cnfor all sufficiently large n, and if the sequences {an}∞n= 1and {cn}∞n= 1bothconverge to the same limit, then the sequence {bn}∞n= 1converges, and to the same limit.Math 409-502 September 17, 2004 — slide #2Limit theorems continuedLocation theoremsIf an→ L and if an< M for all sufficiently large n, then L ≤ M.The example an= n/(n + 1) shows that we cannot draw the conclusion L < M.If an→ L and L < M, then an< M for all sufficiently large n.The example an= (n + 1)/n shows that the hypothesis L ≤ M is insufficient.Math 409-502 September 17, 2004 — slide #3SubsequencesA non-convergent sequence may have convergent subsequences.Example:14,34, 2,18,78, 4,116,1516, 8,132,3132, 16, . . .If a sequence converges, however, then every subsequence converges to the same limit.Example:13,15,17,111,113,117,119, . . . is a subsequence of the convergent sequence {1n}∞n= 1, so itconverges to 0.Math 409-502 September 17, 2004 — slide #4Homework1. Read sections 5.4 and 5.5, pages 68–73.2. Do Problem 5-7, page 75.Math 409-502 September 17, 2004 — slide
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