Originally Posted by

**MathMarshall** I got this "proof" that showed 1+2+3+4+...=-1/12 from a knowledgeable friend about one and a half months ago now. On first glance, I dismissed his proof as probably wrong as the proof seemed counterintuitive and not in line with how we would expect a diverging series to behave. Here goes.

$\displaystyle s= 1+2+3+4+... $

$\displaystyle s_{1}=1-1+1-1+1-1...$

$\displaystyle 2s_{1}=(1-1+1-1+1-1...)+(1-1+1-1+1-1...)$

$\displaystyle 2s_{1}=1$

$\displaystyle s_{1}=1 \div 2=0.5$

$\displaystyle s_{2}=1-2+3-4+5-6+7-8+9...$

$\displaystyle 2s_{2}=(1-2+3-4+5-6+7-8+9-...)+(1-2+3-4+5-6+7-8+...)$

$\displaystyle 2s_{2}=1-1+1-1+1-1+1-1+1-1....$

$\displaystyle 2s_{2}=s_[1]=0.5$

$\displaystyle s_{2}=0.5 \div 2= 0.25$

$\displaystyle s-4s=s_{2}$

$\displaystyle (1+2+3+4+5+6+...)-(4+8+12+...)=1-2+3-4+5-6+7-8+...$

$\displaystyle s-4s=s_{2}$

$\displaystyle -3s=s_{2}=0.25$

$\displaystyle s=0.25 \div -3 =-1/12$

I have asked around from my seniors(I am in Year 1 of a "high school" that starts at 13 and continues for 6 straight years until 18) but I still am not sure what is wrong with the proof apart from some possible notation errors. What is wrong with his "proof" or is he actually correct? s and all its other subscript variants are variables. The first mention of one such variable defines it.