Is there an intuitive way of thinking why a rearranged conditionally convergent series yields different results?

Question

We know that any conditionally convergent series can be made to converge to anything, or even diverge. My question is if we can intuitevely explain why such a thing happens. One would be tempted to think that "eventually, we're summing the same terms". This is wrong, however, because there is no "eventually" in a series.

How can we think about this then?

Answer 1

That the series is not absolutely convergent means that "there's always enough left to go as far as you want". That it's also convergent means that "there's always enough left to go as far as you want in either direction" (because otherwise, if you could go as far as you want in one direction but not the other, the series would go off in that direction and not converge). Now, if there's always enough left to go as far as you want in either direction, you can pick an arbitrary target and then keep going past it to either side by using the appropriate terms of the series. On the other hand, since the convergence implies that the terms go to zero, you need to overstep the target less and less, allowing the rearranged series to converge to the target. That the terms go to zero also ensures that you can use all of them eventually (otherwise you might get a rearranged subsequence but not a rearrangement of the entire series).

Answer 2

Take the classic series discovered by Riemann that converges to $ln(2)$ on this Wikipedia page.

The two "components" of the series are the sums $$\sum^{\infty}_{k=0}\frac{1}{2k-1}$$ and $$-\sum^{\infty}_{k=0}\frac{1}{2k}$$ Now, the first sums to $+\infty$ and the second to $-\infty$, so by rearranging the terms of this series we can get it to converge to any number - for example, if you wanted the series to converge to a larger number than $ln(2)$, then since the positive component of the sum diverges, you always have enough "left" to "get the sum up" to that number and then converge. Similarly, if you wanted it to converge to a negative number, then since the negative component diverges, you always have enough negative terms left to push the sum down to that desired negative number and to then converge to it.

Hope this helps!