Let $a_n\ge 0.$ If $\sum a_n$ converges, say to 1 (such as a probability mass function), then by the Divergence Test we have $\lim_n a_n=0. $ However, can we say how fast it converges?
I was thinking of using the integral test but that only applies for decreasing sequences.
On
No, you can't. Think about the series, $C(k)=\sum_n\frac{1}{n^k}$, $C(k)>1$ are some constants. Then $\sum_n\frac{1}{C(k)n^k}=1$, but the decay rate of $\frac{1}{C(k)n^k}$ varies for different $k$.
Also, consider sequence $\{1,0,0,\cdots\}$ as JMoravitz mentioned in the comment, where already $a_k=0$ for $k>1$.
The answer is no. We cannot say anything about the speed of convergence. We can only say that $a_n\to0$ as $n\to\infty$. This is a question about the nonexistence of the boundary between convergent and divergent series (see this question for more details).