Thank you for the reply, but showing that the limit does not exist does not prove that the sequence diverges. I actually know that it converges and I know what it converges to, but I don't understand how to prove it mathematically.
I constructed this problem intentionally so that it would converge while the limit does not exist, but I made a bad assumption about limits and basically got lucky. It converges to -1/2, but I don't know how to show this.
Surely you know that does not exist,
If you have doubts about that then you have no business trying this question.
If you do understand that fact, then note that
Whatever, if I'm wrong, then explain why it appears that converges to -1/2.
I don't see why it's so hard for you to understand what I'm asking. It is clear that is approaching -1/2 based on the data provided above, so perhaps you should come back and give the problem more than a second of thought.