homework4mat25

homework4mat25 - < a< b Prove b n 1-a n 1< n 1 b n...

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Homework 4 due Thursday August 28th 1. For each of the following sequences ( s n ), find the set S of subsequential limits, and use this to find lim inf s n and lim sup s n . (a) (11.1) s n = 3 + 2( - 1) n . (b) s n = n 2 ( - 1 + ( - 1) n ). (c) ( s n ) = (0 , 1 , 2 , 0 , 1 , 3 , 0 , 1 , 4 ,... ). 2. Prove that every unbounded sequence contains a monotone subsequence that diverges to either or -∞ . 3. Suppose x > 1. Prove that lim x 1 /n = 1. 4. Prove that lim sup( s n + t n ) lim sup s n + lim sup t n . Find an example which shows they are not equal. 5. (12.2) Prove that lim sup | s n | = 0 iff lim s n = 0. 6. In this exercise you’ll prove that the sequence ( s n ) defined by s n = ± 1 + 1 n ² n for n 1 converges. Its limit is the number e , which is approximately 2 . 71828. (a) Prove by induction b n +1 - a n +1 = ( b - a )( b n + b n - 1 a + ··· ba n - 1 + a n ). (b) Suppose 0
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Unformatted text preview: < a < b . Prove b n +1-a n +1 < ( n + 1) b n ( b-a ). (c) Subsitute a = 1 + 1 n +1 and b = 1 + 1 n into the formula in part (b). Use this to show the sequence ( s n ) is increasing, i.e. s n +1 > s n for all n ≥ 1. (d) Prove that the subsequence of even terms is bounded by 4, i.e. ± 1 + 1 2 n ² 2 n ≤ 4 . Do this by plugging a = 1 and b = 1 + 1 2 n into the formula from part (b). Argue that since the sequence is increasing all terms must be bounded by 4. (e) Conclude the sequence converges, and denote its limit by e . (f) Find the limits of ± 1 + 1 2 n ² 2 n , ± 1 + 1 n ² 2 n , and ± 1 + 1 2 n ² n . 1...
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