On the Convergence Rate of a Recursively Defined Sequence

Consider the following recursively defined sequence:
$$ \tau_1 =1,\qquad \sum^n_{j=1} \frac{1}{\sum^n_{s=j}\tau_s}=1\quad \text{for}\quad n\geq 2, $$
which originates from a heat conduction problem first studied by Myshkis (1997). Chang, Chow, and Wang (2003) proved that
$$ \tau_n = \log n +O(1) \qquad \text{for large}\quad n. $$
In this note, we refine this result to
$$ \tau_n= \log n + \gamma+O \biggl(\frac{1}{\log n}\biggr). $$
where $\gamma$ is the Euler constant.