# 1+2+3+4... to infinity

I watched a video where apparently the sum of all natural numbers = -1/12. The video starts by saying

S = 1-1+1-1+1-1+1-1... to infinity. He then says this sum does not have an answer, it's constantly between 1 and 0 depending on where you stop it. So he just takes the average and says 1/2. How is this legit mathematics?

6/0 does not have an answer so we can't just make one up and say oh let's just say it's 94 or something. The thing is it does actually = -1/12 but only if you assume the first part actually does = 1/2, which it doesn't, cos it doesn't have an answer -_-

Samy_A
Homework Helper
S = 1-1+1-1+1-1+1-1... to infinity. He then says this sum does not have an answer, it's constantly between 1 and 0 depending on where you stop it. So he just takes the average and says 1/2. How is this legit mathematics?
Well, it all depends on definitions.
With the usual definition for convergence of series, saying that 1-1+1-1+1- ... equals 1/2 is of course nonsense.

However, using the Cesàro summation, it is correct.

S.G. Janssens
S.G. Janssens
Well, it all depends on definitions.
With the usual definition for convergence of series, saying that 1-1+1-1+1- ... equals 1/2 is of course nonsense.

However, using the Cesàro summation, it is correct.
Exactly. A Belgian beat me to it.

Samy_A
1 + 2 + 3 + ... does not equal -1/12. In fact, this is not a convergent series. (In other words its partial sums 1, 1+2, 1+2+3, 1+2+3+4,... do not approach a limit.)

However, there are ways of taking the sequence of terms (1, 2, 3, ...) and using a "reasonable" method to come up with a real number.

These are (finally!) discussed nicely in the Wikipedia article on the subject.

For instance, one could notice that 1 + 2 + 3 + ... is just like the series

1 + 1/2s + 1/3s + ...,​

which in fact does converge for s > 1, and is known as the (Dirichlet series for the) zeta function ζ(s) of s, when s is set equal to -1. Using the technique called analytic continuation, the zeta function
ζ(s) is analytic on the entire complex plane ℂ with the sole exception of at the value s = 1, where it is not defined (because it has a pole).

Specifically, 1 + 2 + 3 + ... is the Dirichlet series for zeta, BUT with the number -1 plugged in for the variable s. Although as we observed this does not converge, you could try an end run around this fact by simply plugging in -1 not to the Dirichlet series but instead to the zeta function. This might be a good time to stop and note the difference between these two things.

Using this last idea, it so happens that at s = -1, the value of the zeta function ζ(-1) = -1/12.

That is how the otherwise nonsensical equation

1 + 2 + 3 + ... = -1/12​

is "justified". (But in reality it is not justified, since it is untrue.)

Similarly:
Let S = 1+2+4+8+16+.....
2S= 2+4+8+16+32+....
So 2S= S-1
S =-1