https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
How 1+2+3+ ... = - 1/12 ? Why? Shouldn't the result be infinite?
2026-03-26 17:28:17.1774546097
How to solve this equality?
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First of all, it is important to note that 1+2+3.. is a divergent series. Ie, it does not directly add up to any value. This is because there is no limit to the sequence of partial sums.
However, there are other ways that we can use to manipulate this sum and come up with other interesting/important results.
There are several different approaches that come up with this result of -1/12, but a common derivation involves the systemic manipulation of a few other infinite sums.
Il demonstrate below,
Let
S= 1 + 2 + 3 + 4 + 5 + 6.....
4S= 4 + 8 + 12+....
Note, S-4S =-3S = 1-2+3-4+.. ( Which can be represented by the power series as 1/(1+x)^2, and with 1 subbed as x we have 1/(1+1)^2= 1/4
Now we have -3S=1/4
S= -1/12
There ya go!
PS. There are much more sophisticated answers to this question as well, that I am not able to explain/ you might not be looking for. But just to point out that there is other significance to this!