"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 $|x-\sum^N x_n|&lt;\varepsilon$. 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 $|\sum^N x_n|&gt;M$. 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 $M\to\infty$.
Think about those two series (we'll call them S1 and S2) you mentioned that converge to the same number. Let say that both series get to that number after at least 50 terms in each series. So, at 50 terms, the sum you have at that point in S1 will be different than the sum of S2 at 50 terms. The series with the sum at 50 terms that is closer to the convergence point is the series that is approaching it faster at 50 terms.