I don't think any of those are right. The latter two are trivially equivalent, the former also is since it differs by a factor of at most 3 for all n >= 1.
I still think the simplest definition for o(f) is 'g in o(g) iff O(g) is a proper subset of O(f)'. Or you can just say g is dominated by f in the sense that lim sup g/f = 0.
I'm fairly sure those two are equivalent, but it's a good little exercise.
I don't think any of those are right. The latter two are trivially equivalent, the former also is since it differs by a factor of at most 3 for all n >= 1.
I still think the simplest definition for o(f) is 'g in o(g) iff O(g) is a proper subset of O(f)'. Or you can just say g is dominated by f in the sense that lim sup g/f = 0.
I'm fairly sure those two are equivalent, but it's a good little exercise.