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pra-140-final

pra-140-final - n →∞ y n 9 Prove that if for a...

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Math 140, Winter Practice problems February, 2010 Instructor: Professor Ni 1. Exercise 8 of Ch4. 2. Exercise 14 of Ch4. 3. Exercise 18 of Ch4. 4. Exercise 19 of Ch4. 5. Find the limit lim x 0 1 + x - 1 - x (1 + x ) 1 / 3 - (1 - x ) 1 / 3 . 6. Assume that the sequence { a n } is convergence and a n > 0. Prove that lim n →∞ ( a 1 · a 2 · ··· · a n ) 1 /n = lim n →∞ a n . 7. Assume that { a n } satisfies that 0 a n + m a n + a m . Prove that { a n n } converges. 8. For any true statement below, prove it. For any false statement below find an example, or prove it false if you prefer. (i) liminf n →∞ x n + lim inf n →∞ y n liminf n →∞ ( x n + y n ) liminf n →∞ x n + lim sup n →∞ y n . (ii)liminf n →∞ x n +lim sup n →∞ y n limsup n →∞ ( x n + y n ) limsup n →∞ x n +limsup n →∞ y n . (iii) Assume that x n ,y n 0. Then lim inf n →∞ x n · liminf n →∞ y n liminf n →∞ ( x n · y n ) liminf n →∞ x n · limsup n →∞ y n . (iv) Assume that x n ,y n 0. Then liminf n →∞ x n · limsup n →∞ y n limsup n →∞ ( x n · y n ) limsup n →∞ x n · limsup
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Unformatted text preview: n →∞ y n . 9. Prove that if for a nonnegative sequence { a n } it holds that for any sequence { b n } , limsup n →∞ ( a n + b n ) = lim sup n →∞ a n + lim sup n →∞ b n and limsup n →∞ ( a n · b n ) = lim sup n →∞ a n · limsup n →∞ b n . then { a n } must converges. 10. For a sequence { a n } with a n > 0, if lim n →∞ a n +1 a n = a then lim n →∞ a 1 n n = a. You need to show that { a 1 n n } converges. 11. Prove that for function f ( x ) defined on [ a, + ∞ ), satisfying that f ( x ) is bounded on any finite ( a,b ). Then lim x →∞ f ( x ) x = lim x →∞ f ( x + 1)-f ( x ) . Here we assume that both limit exist. You only need to show that they are the same. 1...
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