mat25-hw6

# mat25-hw6 - (a) lim n 1 n 3 (b) lim n 3 n 2-1 n + 1 (c) lim...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Homework Assignment #6 Homework due: Tuesday 5/10/11 at beginning of discussion section Reading material. Read section 2.8 in the textbook. Problems 1. Let ( s n ) n =1 and ( t n ) n =1 be sequences that converge to limits S = lim n →∞ s n and T = lim n →∞ t n . We proved in class a theorem that says that the sequence s n + t n converges to the limit S + T . By imitating the proof of that result, show that the sequence s n - t n converges to S - T . 2. Let ( s n ) n =1 and ( t n ) n =1 be deﬁned by s n = n + 1 , t n = n. Explain why the result in problem 1 above does not apply. Prove however that lim n →∞ ( s n - t n ) = 0 . Hint: Use the identity a - b = a - b a + b and the squeeze theorem. 3. Compute the following limits. For each computation, provide a brief explanation how you obtained your answer from known results.
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Unformatted text preview: (a) lim n 1 n 3 (b) lim n 3 n 2-1 n + 1 (c) lim n 3 n 2 + sin( n ) n 2 (d) lim n 5 n 4-2 n 2 + n + 1 n 2 ( n 2 + 1) 4. Let ( s n ) n =1 and ( t n ) n =1 be sequences. Decide which of the following statements are true. In each case, provide a proof or a counterexample. (a) If ( s n ) n =1 and ( t n ) n =1 are both divergent then so is ( s n + t n ) n =1 . (b) If ( s n ) n =1 and ( t n ) n =1 are both divergent then so is ( s n t n ) n =1 . (c) If ( s n ) n =1 and ( s n + t n ) n =1 are both divergent then so is ( t n ) n =1 . (d) If ( s n ) n =1 is convergent then so is (1 /s n ) n =1 . 1...
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