sa2 - X n =1 1 n 2 = 2 6 . Evaluate X n =1 1 (2 n-1) 2 ,...

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Self-Assessment Questions – Math 118A, Fall 2009 1. Evaluate the limit lim n →∞ ± n - n e ² 1 + 1 n ³ n ´ . without using L’Hospital’s rule. 2. Prove that lim n →∞ ² 1 + 1 n ³ n exists using the following steps: (a) Prove that a n = ² 1 + 1 n ³ n and b n = ² 1 - 1 n ³ n are both increasing sequences. For this you might want to use the fact that m s 1 . . . s m . s 1 + ··· + s m m . (b) Prove that c n = ² 1 + 1 n ³ n +1 is a decreasing sequence. (c) Prove that a n c n , and prove that the limit exists. 3. Let { F n } be the Fibonnaci sequence, i.e., F 1 = 1; F 2 = 1; F n +1 = F n + F n - 1 n 2 . Prove that the series X n =1 F n 3 n converges, and evaluate it. 1
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2 4. Take as a given that
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Unformatted text preview: X n =1 1 n 2 = 2 6 . Evaluate X n =1 1 (2 n-1) 2 , and X n =1 (-1) n 1 n 2 . 5. Call a mapping of X into Y open if f ( V ) is an open set in Y whenever X is an open set in X . Prove that every continuous open mapping from R into R is monotonic. 6. Consider a sequence of positive numbers { a n } such that a n + m a n + a m n, m . Prove that lim n a n n exists. 7. Evaluate the following limit directly: lim x a x n-a n x-a , for n N ....
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sa2 - X n =1 1 n 2 = 2 6 . Evaluate X n =1 1 (2 n-1) 2 ,...

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