Sunday 29 October 2017

sequences and series - Is the sum of natural numbers equal to $-frac{1}{8}$?

I came across the following video on YouTube: Sum of all natural numbers (- 1/8).



Basically what happens is:
\begin{align*}
1+2+3+\dotsb &= N \\
1+(2+3+4)+(5+6+7)+\dotsb &= N\\
1+9+18+27+\dotsb &= N\\
1+9(1+2+3+4+\dotsb)&= N\\

1+9N &= N
\end{align*}
and therefore $N=-\frac{1}{8}$.



This is directly in contradiction with the well-known result of $-\frac{1}{12}$.



What is the problem with this reasoning? Was this result discovered earlier? Is this a consequence of Riemann's Rearrangement Theorem? Thanks in advance.



This was a repost of my previous post because some people said it was a duplicate to "Why is the sum of natural numbers $-1/12$?"

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