So, the problem that we were solving was
To figure out whether the series converged or diverged, after simplification, I asked my professor whether finding the limit of the inside of the function to determine whether the inside function was divergent or convergent would be helpful. My logic was this: If it's divergent, infinity to the power of $n$ as $n$ approaches infinity is just infinity, and if it's between $-1$ and $1$ then it approaches a finite value, right? Meaning that, if we were to take the limit of the inside it would ultimately determine what the function did after taken to the power of n.
I was told that we simply couldn't do this but her explanation was a bit lackluster, it was basically "because I said so." Please tell me exactly how I'm wrong so I can better understand what I'm doing. Thank you!