Sum of all Natural Numbers = -1/12 !?

Quick post about something I came across the other day. I’m currently learning the Principle of Mathematical Induction(PMI) in MATH 220. It is essentially a way of proving a statement is true for all natural numbers. It has a lot to do with proving the summation of sets. However, I will talk more about the PMI in a different post.

A math meme page I follow on Facebook posted:

Image may contain: 4 people, people smiling

It’s a joke about the pineapple pen guy but in math terms. The meme is making fun of the theory/proof that the sum of all natural numbers converges or equals to -1/12 or 1+2+3+4+…+∞ = -1/12. I was really confused and I even believed it at the time! (it is obviously not true though)

I googled it and one of the most popular result is this video which “proves” the theory:

I won’t go too in depth but the proof is incorrect due to it assuming many things that are not true. For starters, the video attempts to assume that 1-1+1-1+1… =1/2. The video does not explain why and assumes that it is common knowledge. However, assume 1-1+1-1+1… is a finite sum Z. Adding Z to itself you would get:
Z+Z=1-1+1-1+1…1-1+1-1+1… but this is just the original sum.
This implies Z+Z=2Z=Z and since Z=1/2, it follows that 1/2 = 1. Which is not possible. Therefore the video’s proof was flawed. This is just one of their incorrect assumptions. For more a more in-depth and thorough analysis, I definitely recommend reading this article:

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