A question on the convergence of series I was reading Protter's "Probability Essentials" and the author says in the fourth chapter that the convergence of series depends on the order in which they are enumerated. I cannot think of a situation in which by changing the enumeration , a converging series might cease to converge or vice-versa. Could someone please give me an example?
Thank you

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...=\sum_{n=1}^\infty (-1)^{n-1} \frac{1}{n}$$
Pick a real number $L$. 
Rearange the series the following way: pick the first positive terms until the sum exceeds $L$. Pick now the first negative terms so that, added to the first sum you get back under $L$.
Repeat: Whenever when above $L$, pick the first left negative terms until the sum is under $L$.  Whenever when below $L$, pick the first left positive terms until the sum is over $L$. 
This is possible because 
$$1+\frac{1}{3}+\frac{1}5+...=\frac{1}{2}+\frac{1}{4}+...=\infty
$$
This can always be done with series which are convergent BUT not absolutely convergent.
