Laplace II: Steps, Impulses & Convolution · Challenge problem

The floor function satisfies $\lfloor t\rfloor=\displaystyle\sum_{k=1}^\infty u(t-k)$ for $t\ge0$. Find $\mathcal{L}\{\lfloor t\rfloor\}(s)$ and evaluate it at $s=\ln 2$.