I recently proved that if $\sum_{n=0}^{\infty} a_n$ converges to some value $S$, then $\sum_{n=0}^{\infty} a_nr^n$ is Abel summable to $S$.
My problem is probably more elementary, as I'm quite new to analysis and have always had a bit of a tough time with power series. I can't seem to find a way to make this sum converge, so I'm not sure how to apply what I proved about Abel summability. I also have the definition of Abel summability:
$$ \text{If} \lim\limits_{r \rightarrow 1^-} \sum_{n=0}^{\infty} a_nr^n = S \text{, then the series} \sum_{n=0}^{\infty} a_n \text{ is Abel summable to } S.$$
I just can't reconcile this definition or the results of my recent proof with this particular series. Changing the sum to $\displaystyle\sum_{n=0}^{\infty} (-1)^{n-1}n$ doesn't really improve the situation. What am I doing wrong here?