new 3n + 11 new i(3n + 11) \"ADO 3 + % lining\for strictly positive integers n. Determine whether this sequence converges, diverges to co, diverges to diverges in some other way. If it converges, nd its limit. (Remember to show your work.) 00 7. (5 points.) The sequence (on) 1 is given by the formula (2n 1): (2n + 1): for strictly positive integers n. Determine whether this sequence converges, diverges to co, diverges to diverges in some other way. If it converges, nd its limit. (Remember to show your work.) 14:: n 00 8. (5 points.) The sequence (on) 1 is given by the formula ,1: _ sin(2n) _ 1 + for strictly positive integers to. Determine whether this sequence converges, diverges to co, diverges to diverges in some other way. If it converges, nd its limit. (Remember to show your work.) an 0C) 9. (5 points.) The sequence (on) 1 is given by the formula n: _ n3 + sin(n) _ 13713 + 3 * cos(n) for strictly positive integers 7).. Determine whether this sequence converges, diverges to co, diverges to diverges in some other way. If it converges, nd its limit. (Remember to show your work.) an 00 10. (5 points.) The sequence (on) 1 is given by the formula ,1: R! an 2 37\" for strictly positive integers to. Determine whether this sequence converges, diverges to co, diverges to diverges in some other way. If it converges, nd its limit. (Remember to show your work.) 00 11. (5 points.) The sequence (on) 1 is given by the formula an: on = n(1)n for strictly positive integers n. Determine whether this sequence is strictly increasing, strictly decreasing, not monotonic. Also determine whether it is bounded. DO 12. (5 points.) The sequence (on) 1 is given by the formula 1 on=n+ n n: for strictly positive integers n. Determine whether this sequence is strictly increasing, strictly decreasing, not monotonic. Also determine whether it is bounded. DO 13. (5 points.) The sequence (on) 1 is given by the formula 72n+1 an n+1 n: for strictly positive integers n. Determine whether this sequence is strictly increasing, strictly decreasing, not monotonic. Also determine whether it is bounded. 14. (5 points.) Let (bn)::1 be a sequence with 5,, > 0 for every strictly positive integer n. limnhm ln(b,,,) : oo. What, if anything, does that say about limnam b\"? 01' 01' GT or OT 0T 0T 01' Suppose 15. (5 points.) Consider the sequence given by an = Vn for strictly positive integers n. You want to prove that limn +co an = co. So, for example, for M = 19 you should be able to find some integer N > 0 such that for all n > N we have an > M. Find such a value of N, and show that it works. (You need not find the best value of N.) (In more colloquial wording, how many terms are needed for the sequence to get and stay bigger than 19?) n2 16. (5 points/part.) Consider the sequence given by an = n2 for strictly positive integers n. (a) Find the limit L of this sequence. (b) Find an integer N that has the property that Jan - L| 0. Your choice of N will, of course, need to depend on E.)