Unformatted text preview: { a 2 n } converges, yet { a n } does not converge. (b) Let a n = 2 /n1 /n 2 + 5. Find the smallest N such that  a n5  < 1 / 100 for n ≥ N . (c) Use an ±N argument to show that a n converges to 5. (7) Show that the interval (1 , 5] is not closed (8) Show that if x n converges to c then so does x n +1 . (9) Let x 1 = 2 and deﬁne x n +1 = 1 6 ( x n + 4). (a) Show that x n +1 < x n is equivalent to x n > 4 / 5. (b) Use induction to show that x n > 4 / 5 and conclude that x n is decreasing. (c) Does x n converge (show why or why not)? If so, ﬁnd the limit....
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 Summer '08
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 Mathematical Induction, Natural number, Prime number

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