Suppose that

$\mu_k$ is an increasing sequence of numbers such that $0 < \mu_1 \leq \mu_2 \leq ..$ with $\mu_k \to \infty$ as $k \to \infty$

$\sum_{k=1}^\infty |u_k|^2 < \infty$ and $\sum_{k=1}^\infty \sqrt{\mu_k}|u_k|^2 < \infty$ where $u_k$ is a given sequence of real numbers

I want to show that (if true) the sum $$\sum_{k=1}^\infty |u_k|^2 \mu_k \frac{\cosh(2\sqrt{\mu_k}(T-t))}{\sinh^2(\sqrt{\mu_k}T)}$$ is uniformly convergent in the variable $t \in [\epsilon, T]$ for $\epsilon > 0$? (in the previous version of this thread I forgot to exclude $t=0$.)

The problem is that the numerator contains an exponential term with the "wrong" sign.

The motivation is, I want to integrate this sum term by term because it gives me a bound on a norm (the sum comes from a solution to an differential equation). Thanks for help.