This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.

Let $a_k >0$ be an increasing sequence of real numbers. Let $f_k$ be real numbers.

Let $R \in (0,\infty)$ and $y \in (0,\infty)$. Let $$v(y) = \sum_{k=1}^\infty f_k\left(B_1(k,R)e^{-a_ky} -B_2(k,R)e^{a_ky}\right)$$ where $$B_1(k,R) = \frac{e^{a_kR}}{e^{a_kR}-e^{-a_kR}}\quad\text{and}\quad B_2(k,R) = \frac{e^{-a_kR}}{e^{a_kR}-e^{-a_kR}}.$$ We also know $\sum_{k=1}^\infty f_k < \infty$.

Is $v(y)$ finite for all $y$ and $R$ (i.e. does the infinite sum exist)? Also I would like to know whether the series is uniformly convergent (so whether the partial sums uniformly converge) so that I can do term by term integration.

I don't see why the sum even exists...

Remark: In fact, $a_k^2$ are the eigenvalues of the Neumann Laplacian, and $f_k := (u,\varphi_k)_{L^2}\varphi_k(x)$ on some bounded domain $\Omega$ where $\varphi_k$ are the eigenvectors of the Neumann Laplacian.