Suppose that 


1. $\mu\_k$ is an increasing sequence of numbers such that $0 < \mu\_1 \leq \mu\_2 \leq ..$ with $\mu\_k \to \infty$ as $k \to \infty$
2. $\sum\_{k=1}^\infty |u\_k|^2 < \infty$ 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.



Is this series involving hyperbolic functions uniformly convergent?
The answer is negative. 


Take $\mu\_{k}=k$, $u\_{k}=\frac{1}{k}$ and $t=0$. Conditions 1 and 2 hold. Notice that $\frac{\cosh(2\sqrt{\mu\_{k}}T)}{\sinh^{2}(\sqrt{\mu\_{k}}T)}=1+o(1)$ and the sum diverges. 


If you want it to be uniformly convergent you need either $\sum |u\_{k}|^{2}\mu\_{k}<\infty$ or $t\in [\varepsilon,T]$ for some $\varepsilon >0$.


