It's time for you to get to the root of the problem. You're ready to rho, rho, rho yourself out of this pickle you're in. If you're in a world where the n's have taken power in your summations, I have a solution for you: The Root Test.
If you want an equestrian analogy, here it is:
Think about a horse going into a stable for the night. The horse being the an and the stable being the nth root.
Now think of the n's inside the root as flies and the n outside the root as fly spray.
The n's in power will cancel, but by the end of the night something will have happened. If you take the limit of that equation you will be left with ρ. If we say that rho stands for the number of ponies left, here is the result:
ρ<1 means you have less than one pony in the morning. That night was lord of the flies and you were a victim. The amount of ponies will converge.
ρ=1 means that you're at a stalemate. The flies didn't really do much of anything and you still have your one pony. The test is inconclusive.
ρ>1 means you have found more than one pony in your stable in the morning. The flies were a blessing and you now have an infinite amount of ponies. The series diverges.