Thread: Help find errors in this "proof" that the limit of 1+2+3+4... = -1/12

1. Help find errors in this "proof" that the limit of 1+2+3+4... = -1/12

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.

2. Re: Help find errors in this "proof" that the limit of 1+2+3+4... = -1/12

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.
I'm hardly the last word on this, but you have an error right at the start. Neither s nor s1 converge. Doing algebra with non-converging sums is a very tricky and likely way above the level of what what you are trying to do.

-Dan

3. Re: Help find errors in this "proof" that the limit of 1+2+3+4... = -1/12

I agree with topspark, but if you are looking to baffle your friends, looking to different number systems are a good place to start. Consider the following sum:

$S(x) = 1+x+x^2+x^3+\cdots$

This is known as a geometric sum.

$xS(x) = x+x^2+x^3+\cdots$

$S(x) - xS(x) = (1+x+x^2+x^3+\cdots ) - (x+x^2+x^3+ \cdots) = 1 + (x+x^2+x^3 + \cdots ) - (x+x^2+x^3+\cdots ) = 1$

$(1-x)S(x) = 1$

$S(x) = \dfrac{1}{1-x}$

Now, this infinite sum converges for any $-1<x<1$. So, for example:

$S\left(\dfrac{1}{2}\right) = 1+\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots = \dfrac{1}{1-\dfrac{1}{2}} = 2$

If you use a $p$-adic number system, this infinite sum converges when $|x|_p < 1$. Without going into too much detail of what a $p$-adic number system is, the infinite sum:

$S(2) = 1+2+2^2+2^3+\cdots = \dfrac{1}{1-2} = -1$

converges in the $2$-adic number system. The math to prove this is true is well above the topic of conversation, but this blew my mind when it was first introduced to me by my 7th grade math teacher to get me to stop disrupting class. Even now that I understand some of the principles of $p$-adic numbers, this still blows my mind.

$1+2+2^2+2^3+\cdots$ does not look like it should ever converge! Many mathematicians are so indoctrinated in the real number system that this could surprise even them!

4. Re: Help find errors in this "proof" that the limit of 1+2+3+4... = -1/12

The equation is in part the basis of String Theory (yes the math is sound, albeit nonsensical)