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:
#SoME1

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Thank you for your support!

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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:

[joaofrancisco]

I'm mesmerized by how intuitive you made the theorem seem. I always felt it was sort of like a "paradox", but your explanation made it look almost plain obvious. Great video!

[tlanohoecr]

My strategies for the extra problems:

To make the series oscillate, instead of having one target sum, make it two, for example 1 and 0. First take enough positive terms to get the partial sum above 1, then take negative terms to get it below 0, then repeat. This way the series will have infinitely many partial sums both above and below the interval [0, 1].

To make the series diverge to infinity, use the same strategy, but make the target sums increase by 1 each time they are reached. I.e. first take positive terms to get above 1, then take negative terms to get below 0, then get above 2, then below 1, then above 3, then below 2, etc. Since the negative terms converge to 0, the sequence of partial sums will have an increasing lower bound that will go to infinity.

[FrostDirt]

Sometimes I get frustrated by how you always state the obvious (you do it very slowly, of course). Then I realise that I would never have come up with what is "obvious" in the middle of the video had you not told me about what is "obvious" previously. And then I just realise that that's how math works! You just state the "obvious" and you come up with more "obvious" statements. This proves how good of an educator you are, great job.

[kalebmark]

I like how this is your only video and it's an absolute banger.

[noahnaugler]

It makes sense to a degree, the same way that ∞ - ∞ can equal whatever you want

[menturinai]

I remember learning about conditional convergent series when taking calculus class and it always felt "why do I care if a series is conditionally vs. absolutely convergent". Your video answered my long-standing question. Thank you!

[systemsbyvedant]

This explanation video was on par with 3b1b, if not better. As a very loyal 3b1b viewer, I want to emphasize that it means a lot.

[MartinPoulter]

A really impressively clear explanation. Thanks a lot for making this!

[debblez]

fun fact: for the (-1)^n/(2n+1) sequence, the series exactly equals arctanh(p)/2 + pi/4 where p is the proportion of negative and positive terms, ranging from -1 (all negative terms) to 1 (all positive terms)
For example, the ++- pattern converges to arctanh(1/3)/2 + pi/4
I would love to see a proof for why this is the case, because it seems too simple not to have an elegant reason for being that way.

[ekut]

this video made me feel like I knew a lot about the subject by simply explaining the topic really well

[clementdato]

A minor mistake in the argument at around 8:52 for about why the added terms must converge to 0. It is not because otherwise it must diverge to infinity, but because otherwise it must diverge to infinity OR by oscillating (e.g 1 -1 1 -1… series does not diverge to infinity.)

[mCoding]

Subscribed and watched every video. Great job with the visualizations and distilling the salient features and ideas of the proof into something not just manageable but intuitive, all without any handwaving. Keep going!

[kumozenya]

I think this one might be my fav one so far! Amazing job! the animation you have made it really to follow along and helped me understand the concept a lot!

[synaestheziac]

This was an incredibly lucid explanation. I think I’m actually going to try to teach this to the students in my math-for-art-students course. Before seeing this video, I wouldn’t have dreamed of trying to explain something like this to them, but if I replicate your explanation, I think some (hopefully most) of them might actually get it!

[PYME]

It is amazing how it has already been two years... I remember the time I watched your first video when it was 2 months old.... time flies by.

[yahav]

amazing. I felt I watched a 5 minute video since your presentation was smooth and intuitive, great job!!

[soupisfornoobs]

I don't know what about your channel is different, but you've helped me understand these paradoxical maths things better than anyone else! Both this and fractional derivatives, finally explained understandably

[rubetz]

❤This is my favorite theorem in the whole calculus, both for sounding so counterintuitive and with so intuitive a proof.

[ZedaZ]

You must have put a whole lot of work into this and it was so good, thanks a bunch for this!

[yifeifu]

What an explanation!!! Absolutely the best Maths video I've seen this year!