sa118A_2

sa118A_2 - Self-Assessment Questions – Math 118A Fall...

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Unformatted text preview: Self-Assessment Questions – Math 118A, Fall 2009 1. Evaluate the limit n lim n − n→∞ e 1 1+ n n . without using L’Hospital’s rule. 2. Prove that n 1 1+ n lim n→∞ exists using the following steps: (a) Prove that n an = 1 1+ n 1− 1 n n bn = and are both increasing sequences. For this you might want to use the fact that √ s1 + · · · + s m m s1 . . . sm . ≤ . m (b) Prove that cn = 1 1+ n n+1 is a decreasing sequence. (c) Prove that an ≤ cn , and prove that the limit exists. 3. Let {Fn } be the Fibonnaci sequence, i.e., F1 = 1; F2 = 2; Fn+1 = Fn + Fn−1 ∀n ≥ 2. Prove that the series ∞ n=1 converges, and evaluate it. 1 Fn 3n 2 4. Take as a given that ∞ π2 1 =. n2 6 n=1 Evaluate ∞ 1 , (2n − 1)2 n=1 and ∞ n=1 (−1)n 1 . n2 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 {an } such that an+m ≤ an + am ∀n, m ≥ 0. Prove that an n→∞ n lim exists. 7. Evaluate the following limit directly: x n − an , x→a x − a lim for n ∈ N. ...
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sa118A_2 - Self-Assessment Questions – Math 118A Fall...

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