This question is an extension of this one, in which I am told that, given a sequence $a_1, a_2, a_3, ...$, $$\sum_{j=1}^{\infty }a_{j}=\sum_{n=1}^{\infty }(\sum_{k=2^{n-1}}^{2^{n}-1}a_{k})$$
is only true when $\sum _{j = 1} ^{\infty} a_j$ converges. This seems counterintuitive to me (though I know sometimes intuition can be a dangerous thing when dealing with infinities). Is there a place I can read more about this, and are there any easy concrete examples of equality not holding?
Think of $(1-1)+(1-1)+(1-1)+\dots=1-(1-1)-(1-1)-\dots$