Does the Abel sum 1 - 1 + 1 - 1 + ... = 1/2 imply $\eta(0)=1/2$? If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a neighborhood of $s=0$ such that $F(0)=A$. Is that true? 
Edit.
The Dirichlet series $\sum_{n=1}^\infty a_n n^{-s}$ converges absolutely for $\Re(s)>1$, because the radius of convergence of the power series $\sum_{n=1}^\infty a_n x^n$ is at least $1$, that is to say
$$
\frac{1}{\lim\sup_{n\to\infty}\sqrt[n]{|a_n|}}\geq1
$$
and, thus, there exists $N\in\mathbb{N}$ such that $|a_n|\leq1$ for all $n\geq N$.
 A: this is an (incomplete) sketch of proof I planned to use at first for answering to The value of a Dirichlet L-function at s=0.
let $F(s) = \sum_{n=1}^\infty a_n n^{-s}$.


*

*suppose it is converging for $Re(s) > 0$ and $F(0) = \lim_{s \to 0^+}F(s)$ exists. $a_n = \mathcal{O}(1)$ and $f(x) = \sum_{k=1}^\infty a_n x^n$ converges for $|x| < 1$. 

*suppose also that $f$ doesn't have a singularity at $s= 1$.


from  $n^{-s} \Gamma(s)  = n^{-s} \int_0^\infty y^{s-1} e^{-y} dy = \int_0^\infty x^{s-1} e^{-nx}dx$ (with the change of variable $y = nx$) and because when $s \to 0^+$ :  $\Gamma(s) \sim \frac{1}{s}$ we get when $s \to 0^+$: 
$$\frac{F(0)}{s} \sim \Gamma(s) F(s) = \sum_{n=1}^\infty a_n n^{-s} \Gamma(s) = \sum_{n=1}^\infty a_n \int_0^\infty x^{s-1} e^{-nx} dx = \int_0^\infty x^{s-1} f(e^{-x})dx\qquad(1)$$
(inverting $\int$ and $\sum$ by dominated convergence). since $|f(e^{-x})|$ is bounded, that last integral converges for $Re(s) > 0$, and for any $\epsilon > 0$ : $\int_\epsilon^\infty  x^{s-1} f(e^{-x})dx$ converges when $s = 0$. hence when $s\to 0^+$, using the continuity of $f(x)$ on $[0,1]$ :
$$\int_0^\infty x^{s-1} f(e^{-x}) dx \sim \int_0^1 x^{s-1} f(e^{-x}) dx \sim f(e^{0}) \int_0^1 x^{s-1}  dx \sim \frac{f(1)}{s}  \qquad(2)$$
hence :
$$(1) \text{ and } (2)\quad\implies \quad F(0) = f(1)$$
from this you should be able to see how the constraints $a_n = \mathcal{O}(1)$ and both $F(0)$ and $f(1)$ exist  can be discarded or not.
