Convergence of a series of vectors in a Banach space Let $\sum_{k=1}^\infty\lambda^{k-1}\boldsymbol{v}_k$ be a series of vectors where $\lambda^{k-1}\in\mathbb{C}$, or $\lambda^{k-1}\in\mathbb{R}$, and the $\boldsymbol{v}_k$ belong to a Banach space.
I think I have been able -please correct me if I am wrong- to prove that if $\sum_{k=1}^\infty|\lambda^{k-1}|\|\boldsymbol{v}_k\|$ converges, then $\sum_{k=1}^\infty\lambda^{k-1}\boldsymbol{v}_k$ does.
Can we define a convergence radius  as we do when $\boldsymbol{v}_k\in\mathbb{C}$, outside of which $\sum_{k=1}^\infty\lambda^{k-1}\boldsymbol{v}_k$ does not exist? If we can, how can it be proved? Thank you so much for any answer!
 A: The statement you say you have been able to prove is correct. In fact, it's very close to a characterization of Banach spaces. A normed linear space $X$ is Banach if and only if every absolutely summable series in convergent, i.e. if every series $\sum_{n=1}^\infty a_n$, $a_n\in X$ with
$$
\sum_{n=1}^\infty \Vert a_n \Vert_X < \infty
$$
converges in $X$.
This isn't exactly your case since you are talking about power series. This is essentially the idea of Banach-space valued analytic functions: see http://planetmath.org/banachspacevaluedanalyticfunctions for definitions and some properties, and Where does the theory of Banach space-valued holomorphic functions differ from the classical treatment? for an interesting question outlining some more features of the theory. They are quite surprisingly similar to analytic functions of a complex variable.
In principle, of course you can define a radius convergence for a Banach-space valued power series: the largest radius $R$ for which the power series converges for all $|\lambda| < R$. A priori, we won't know for a given power series if this $R$ is positive - just like for the usual real and complex power series. We also won't know in general about convergence outside the disc of convergence, which is also the same for real and complex power series. (You can't say anything about convergence at the boundary of the disc of convergence without having a specific power series in mind.)
