elementary number theory - What can be said about the convergence of this series?

What can be said about the convergence of the following modification of the hyperharmonic series ($\sum_{n=1}^{\infty} \frac{1}{n^{s}}$, which is convergent for any s>1):
$$\sum \frac{1}{n^{s_n}}$$ with $s_n$ strictly monotonically approaching 1 from above? In case both convergence and divergence are still possible under this condition, is it possible to give a specific criteria for convergence, e.g. in terms of the rate of convergence of $s_n$?