"Slow" and "fast" rates of convergence I have recently read about convergence and divergence. However, I am having trouble understanding how something can converge/diverge "slowly" or "fast". If you sum up two series (that converge to the same number) infinitely, they will converge, not at any particular rate, but just to the number- this is how I see it.
So what does it mean for a series to converge (or diverge) slowly?

Suppose $\sum x_n$ converges. Given a positive $\varepsilon$; let $S:(0,\infty)\to \Bbb N$, $S(\varepsilon)$ give the smallest $N$ such that $|x-\sum^N x_n|<\varepsilon$. This measures convergence, in some sense.
Suppose $\sum x_n$ diverges to positive infinity. Given a positive $M$, let $S:(0,\infty)\to \Bbb N$, $S(M)$ give the smallest $N$ such that $|\sum^N x_n|>M$. This measures divergence, in some sense.
As an example, if $x_n=1/n$, then $S(N)\sim e^N$, this says that the divergence is very slow. That is, the larger $S(N)$ is, the slower the series diverges.
For convergence, we're worried at what happens with $S(\varepsilon)$ as $\varepsilon\to 0$. For $S(M)$; we're interested at what happens as $M\to\infty$.
