Let $$f(t) = \sum_{k=0}^\infty ke^{-t\sqrt{k}}u_k$$ for $t \in (0,\infty)$, where the $u_k$ is such that $\sum \sqrt{k}u_k$ converges, but we know nothing about the convergence of $\sum ku_k$.
How do I show that the partial sums that make $f(t)$ converge uniformly to $f(t)$ for $t \in (0, \infty)$?
I can't apply Abel's test since it is not monotone.
I don't think this is true. For example, take $u_{k}=\frac{(-1)^{k}}{k^{3/4}}$.
Of course, you can show that the series converges uniformly on subsets of $R^{+}$ of the form $[a,\infty )$ if you write
$f(t) = \sum_{k=0}^\infty ke^{-t\sqrt{k}}u_k=f(t) = \sum_{k=0}^\infty \sqrt{k}u_k(\sqrt{k}e^{-t\sqrt{k}})$
and use the Weierstrass M-test.