Another way to put the same answer is that if you write the system in polar coordinates (the motion will be in a plane, it's a central force), and just focus on the radial coordinate, conservation of angular momentum allows you to write the situation as if there was an effective repulsive potential that is proportional to the square of the angular momentum, and falls off with distance like the inverse square of r. (For example, see https://en.wikipedia.org/wiki/Effective_potential) Hence, for any real attractive potential that rises less rapidly than inverse square as the particle approaches the center (so n<2), the repulsive potential will eventually win out, and prevent arrival at the center. But if the attractive potential rises more rapidly, as for n>2, then there is no centrifugal barrier, and the potential will cause a monotonic falling in r. The question is then if it reaches the center in finite time, and it will because eventually v(r) will look like a r to a negative power, which does arrive in finite time.
There are several twists that remain, however. If the particle starts out with enough speed to escape the system, then if its angular momentum is large enough, it can avoid the center even if n>2. Also, if n=2, the combined effective potential is flat, so particles initially going inward will hit the center, but particles initially going outward with go to infinity.
Finally, it should be noted that the above expression for the effective potential assumes the speeds are never anywhere close to the speed of light. That will certainly not be true for a particle that falls to the center if n>2, but when relativity enters, it only weakens the centrifugal barrier, so if the particle is going to hit the center in the nonrelativistic analysis, it will in the relativistic one too. However, the situation gets tricky for 1 < n < 2, because then you might think you will not hit the center nonrelativistically, but relativistically, you do hit the center. I presume the question you are asking is not interested in how relativity can complicate the situation, and I think that's only an issue for 1 < n < 2.