Given the set of all power series with radius of convergence ($r$ in the definition) equal to one:
$A:=\{\sum a_kz^k | r =1\}$
Does $A$ form a vector space?
The radius of convergence doesn't change when you multiply a scalar inside, however if you add them the radius will be at least $1$ or bigger than $1$. Also the zero vector would be $a_k=0$ with $k\in \mathbb{N}$ and that would mean the series converges everywhere, so $r=\infty$.
Can we change the conditions of the set so that it becomes a vector space?
Yes, if we set $A:=\{\sum a_k z^k|r\ge 1\}$.
Are these thoughts correct?