Let {a_{n}} be a real positive sequence. Suppose lim n-infinity (a_{n+1})/a_{n} = L. Prove the following: a. If L 1, then {a_{n}} diverges AND is unbounded c. Give examples to show that if L = 1, then {a_{n}} may or may not converge

Question

Let {a_{n}} be a real positive sequence. Suppose lim n->infinity (a_{n+1})/a_{n} = L. Prove the following:
a. If L < 1, then {a_{n}} converges to 0
b. If L > 1, then {a_{n}} diverges AND is unbounded
c. Give examples to show that if L = 1, then {a_{n}} may or may not converge

anonymous · Answer

Honestly, not quite sure how to go about this given the theorems we're allowed to use. I think that I'm allowed to say that if {a_{n+1}}/a_{n} < 1 for part a, I can say a_{n+1} is positive and that a_{n} is decreasing, right? I'm not sure if that's useful, though, you could have a decreasing function with a limit between 0 and 1. Is there a way to use the whole epsilon neighborhood idea? I have a bunch of random ideas written out and none lead me anywhere, lol.