# 6 (Chapter 4)(20 points, Due on Wednesday, October 26) Question 1. [3 points each] Use the Monotone Convergence Theorem to prove that each of the following sequences is convergent and then find the limit of each sequence. (A) s1 3 and s n1 10s n 17 (B) s1 5 and s n 1 s1 2 7 for n 1 . for n 1 . Question 2. [2 points each] Determine whether the given limits exist and find their values. Give clear explanation. (A) lim n 5n 7 2 n1 (B) lim n ( n n n) 2 (C) 1 2 lim n (1 ) n n Question 3. [2 points each] Prove or disprove 1. 2. 3. 4. Every monotone sequence is convergent. Every Cauchy sequence is monotone. Every Cauchy sequence is bounded. Every sequence has a convergent subsequence