The question was:
Guess the limit of the series, and prove your guess from the definition of limits (without any of the theorems thought in class).
I guessed that the limit is $0$, and then looking at the definition of a limit I have: $\forall\epsilon>0\:\exists n_{0}>0\:\forall n>n_{0},\left|a_{n}-0\right|<\epsilon$
This means that I need to find $n_0$ such that for all $n>n_0$, $\frac{n_0}{2^{n_0}-1}<\epsilon$. But since I already prove that the sequence is decreasing, it's enough to find such $n_0$. But... that's where I'm stuck...