Sums of Landau terms revisited
Looks right to me in the following convention:
$S_n = \sum_{k=1}^{n} \Theta(1/k)$ is convenient notation for

There is an $f(x) \in \Theta(1/x)$ (as $x \to \infty$) such that
$S_n = \sum_{k=1}^{n} f(k)$.

Thus the $c_i$ (or with the notation in this answer $c_k$) you get, are not really dependent on $k$.
Under this interpretation, it is indeed true that $S_n = \Theta(H_n)$.
In fact, in Jeff's answer, he shows that $T(k+1) = f(k) + T(k)$ where $f \in \Theta(1/k)$, so it is consistent with the above interpretation.
The confusion seems to be arising from mentally &quot;unrolling&quot; the $\sum$ and presuming different functions for each occurrence of $\Theta$...