One of the most, conceptually speaking, for me to understand is the topic of infinite series. I have always had a hard time proving that an infinite series diverges or even finding a solution for the sum of infinite series. So when I came across a video by Mathologer titled "Riemann's paradox: pi = infinity minus infinity", my mind had a mini-meltdown. In the video he mentions Reimann's Rearrangement Theorem, whereby the sum of the infinite series $1- \frac{1}{2}+\frac{1}{1}-\frac{1}{4}+…$ can equal whatever you want it to, depending on how you group the terms.
Even though in the video he explains how that is possible, I cannot for the life of me understand how different groupings lead to different answers. All my life I thought that no matter how you arrange the terms, because the series is infinite the sum will eventually be the same – no matter how you group them. Obviously, that's wrong but how can I understand this phenomenon better?
Best Answer
The intuitive reasoning is this: if a (real) series is convergent, but not absolutely so, then both the series consisting only of its negative terms and the series consisting only of its positive terms must diverge. If only one of them diverged, then the whole thing would have to diverge, too, and if none of them diverged, then the series of the absolute values would just converge to the sum of the absolute values of the two "subseries".
Now choose whatever value $x$ you want the rearrangement to converge to. For this, take the first few positive terms until you're above $x$. Then take the first few negative terms until you are below $x$. Take the next few positive terms until you are again above $x$, then the next few negative terms until you're again below $x$. Repeat ad infinitum, so you kinda oscillate around $x$, but getting closer to $x$ with every swing. This works because the negative and positive parts diverge, so you can always find enough terms to get you above or below the treshold, but the series itself converges, so the terms get small enough that you "overshoot" less and less with each swing.