Can you change a sum by rearranging its numbers? --- The Riemann Series Theorem

Normally when you add up numbers, the order you do so doesn't matter and you get the same sum regardless. And, of course, the same holds true even if you add up infinitely many numbers.....
Right?

=Chapters=
0:00 - Let's rearrange a sum!
1:48 - Investigation
6:32 - Riemann Series Theorem explained visually
13:58 - Resolving objections
18:52 - A step further and a challenge
20:07 - Significance of the Riemann Series Theorem
21:47 - Final thoughts

This video is a participant in the 3Blue1Brown First Summer of Math Exposition (SoME1). You can find out more about it here:
https://www.3blue1brown.com/blog/some1
#SoME1


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The animations in this video were mostly made with a homemade Python library called "Morpho".
If you want to play with it, you can find it here:
https://github.com/morpho-matters/mor...