In calculus, we learned how to identify whether a series is convergent or not. However, for a divergent series, we never learned how to obtain how fast it diverges, i.e., how to obtain $d$ defined as follows:
$$\lim_{N\to \infty} \frac{\sum_{n=1}^N a_n}{N^d}=c,$$ where $a_n\ge 0$ and $c$ is a non-zero constant.
For cases where the sum of the first $N$ terms can be explicitly calculated, find $d$ is easily. For example, for the series $\sum_{n=1}^N n$, we know $d=2$.
I wonder whether there is a general procedure to obtain $d$ analytically for all divergent series. For example, $$\lim_{N\to \infty} \frac{\sum_{n=1}^N 1/n}{N^d}=c,$$