the dividing of a_n+1 /a_n is not conlusive
Or are you saying that you know you have to prove one of those but don't know how?
I would look at .
Now, in the limit, as k goes to infinity, that fraction goes to 1 so this "eventually" decreases to 0.
If you really need to prove that it is decreasing for all k greater than or equal to 1, I recommend Plato's suggestion. Take the dervative of .