"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?
 A: 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 $M\to\infty$.
A: We define the sum of an infinite sequence $a_1,a_2,\dots$ to be the limit of its partial sums:
$$
\sum_{k=1}^\infty a_k = \lim_{N\to\infty}\sum_{k=1}^N a_k.
$$
The series converges quickly if the limit of partial sums converges quickly; the series converges slowly if the limit converges slowly. Similarly, the series diverges quickly if the limit goes to infinity very quickly and diverges slowly if the limit goes to infinity very slowly.
In the interest of being more concrete -- I never said what I meant by "slowly" or "quickly", did I? -- let me give you a few examples.


*

*The divergent series $\sum_{k=1}^\infty 1$ has partial sum $n$; it diverges as quickly as $n$ grows.

*The divergent series $\sum_{k=1}^\infty 1/k$ has partial sum very close to $\log n$, meaning that it diverges very slowly. To reach a sum greater than $100$, for example, you would need to add up something like the first $3\times10^{43}$ terms.

*The convergent series $\sum_{k=0}^\infty (-1)^k/k!$, which sums to $e^{-1}$, converges quickly: the difference between the $n$th partial sum and the sum of the series is no more than $1/(n+1)!$. This follows from basic properties of alternating series. To calculate $e^{-1}$ to $5$ decimal places you would have to add the first $8$ terms of the series.

*The convergent series $\sum_{k=1}^\infty (-1)^k/k$, which sums to $\ln 2$, converges much more slowly: to calculate $\ln 2$ to $5$ decimal places you would need to add the first $500000$ terms of the series.

A: 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.
