Unformatted text preview: Ç«N proof that lim n â†’âˆž 1 n +1 = 0. * Note: This is book problem 2b. 2. Using only the AP give a direct Ç«N veriÂ±cation that b 2 âˆš n + 1 n + 3 B converges. ** Note: This is book problem 3a. 3. For the sequence { a n } deÂ±ned in book Example 2.3 show that âˆ€ x âˆˆ Q âˆ© (0 , 1] there are inÂ±nitely ** many indices n such that a n = x . ClariÂ±y what this is saying with a speciÂ±c example. Note: This is book problem 5 plus a bit more. 4. Suppose that { a n } converges to a > 0. Show âˆƒ N st n â‰¥ N â†’ a n > 0. * Note: This is book problem 6. 5. DeÂ±ne a 1 = 1 and a n +1 = Â± a n + 1 n if a 2 n â‰¤ 2 a n âˆ’ 1 n if a 2 n > 2 (a) Write the Â±rst Â±ve terms of this sequence. * (b) Show that âˆ€ n ,  a n âˆ’ âˆš 2  < 2 n . ** (c) Use this property to show the sequence converges to âˆš 2. * Note: This is book problem 12 broken up plus a bit more....
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 Spring '08
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 Math, AP, UCI race classifications, Geometric progression, Book problem

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