Actually I think it is easy to show the uniform convergence by splitting the sum into two and using Abel's uniform convergence test (and then recombining the sums).

Hachino, aren't you assuming that $\sum |f_k| < \infty$ in your comment? The $f_k$ are not necessarily positive so I don't see how you can bound it like that to get the uniform convergence (related math.stackexchange.com/questions/1244956/…).

Thanks for this answer. You are right, in fact $y \in (0,R)$, I was mistaken in my OP. One question: if I want to send $R$ off to infinty in the infinite sum denoted by $v(y)$ in my post, can I interchange the limits $\lim_{R}\sum_{1}^\infty (\cdot) = \sum_1^\infty \lim_{R}(\cdot)$? I need uniform convergence wrt. $R$, but I don't know how.