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

## Main Question or Discussion Point

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 -_-

## Answers and Replies

Samy_A
Science Advisor
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
Science Advisor
Education Advisor
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.

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