I often get requests to give a visual proof of the claim that
1+2+3+4+5+... = -1/12
that is, that the sum of all positive integers is -1/12. One problem with this is that it isn't true using the techniques of real analysis, so visualizing it can be challenging (not to mention that visualizing negative numbers is challenging in and of itself). Another challenge is that people get pretty angry with the various methods used to produce this claim.
However, we are able to make sense of this sum using a series of visual arguments that connect three different infinite sums and choose one value to assign to them. Here I try to show visual representations for this argument. While this argument has its inconsistencies, there are more legitimate reasons out there to indicate these values make some sense. In each case, there are valid reasons to assign these values, though the ones here aren't always the best.
This sum has appeared many times on YouTube and has created a lot of controversy and excitement. I am not claiming this to be a set fact; instead, my intent is to show how one standard (though with problems) argument for this claim can be made visual in some sense. If you want to know more about the intricacies involved in this argument, see the videos linked below, especially the ones from Mathologer.
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/visualproofs Thanks!
If you want to find other nice commentaries about this infinite sum, check out the following.
From @numberphile
• ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... =...
• Sum of Natural Numbers (second proof ...
• Why -1/12 is a gold nugget
• The Return of -1/12 - Numberphile
• Does -1/12 Protect Us From Infinity? ...
• One minus one plus one minus one - Nu...
From @Mathologer :
• Ramanujan: Making sense of 1+2+3+... ...
• Numberphile v. Math: the truth about ...
From @blackpenredpen
• Not -1/12
• the most famous Ramanujan sum 1+2+3+....
The alternating geometric series argument comes from a proof by The Viewpoints 2000 group from the October 2001 issue of Mathematics Magazine page 320 (www.jstor.org/stable/2691106 ).
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Mathematical Visual Proofs