What is the Abel sum of this series and why?

Question

$\sum_{n=0}{a_n}$ is Abel summable to L if $f(x) = \sum_{n=0}{a_nx^n}$ converges on $[0,1)$ and $\lim_{x\rightarrow 1-} f(x) = L$.

I know $\sum_{n=0}{(-1)^nx^n}$ is convergent on that interval but what is its Abel sum?

Help appreciated.

Answer 1

Assuming that the sum starts when $n=0$, the answer is $\frac{1}{2}$, because$$\lim_{x\to1^-}\sum_{n=0}^\infty(-1)^{n}x^n=\lim_{x\to1^-}\sum_{n=0}^\infty(-x)^n=\lim_{x\to1^-}\frac1{1+x}=\frac12.$$