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hw3.3 - Math 310 HW 3 Due October 7 2008 Solution...

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Math 310, HW 3, Due October 7, 2008, Solution Instructor: Swatee Naik 1. 3.2: 10 Determine the following limits. (a) lim ( (3 n ) (1 / 2 n ) ) . Note that (3 n ) (1 / 2 n ) = ( 3) (1 /n ) n (1 / 4 n ) . As shown in Example 3.1.11, (c), lim ( ( 3) (1 /n ) ) = 1 . Since 1 n (1 / 4 n ) n (1 /n ) , for every n N , and by Example 3.1.11, (d), lim ( n (1 /n ) ) = 1 , by the Squeeze theorem, we have lim ( n (1 / 4 n ) ) = 1 . Theorem 3.2.3, (a) states that the product of two convergent sequences is convergent and the limit of the product is product of the limits. Therefore, lim ( (3 n ) (1 / 2 n ) ) = 1 . (b) lim ( ( n + 1) 1 / (ln( n +1) ) Note that since n + 1 = e ln( n +1) , we have x n = ( n + 1) 1 / (ln( n +1) = e for all n N . So this is the consant sequence ( e ) which converges to e . | x n - e | = 0 < ± , for all n N and any given ± > 0 . 2. 3.3: 2 Let x 1 > 2 and x n +1 := 2 - 1 /x n for n N . Show that ( x n ) is bounded and monotone. Find the limit.
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hw3.3 - Math 310 HW 3 Due October 7 2008 Solution...

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