When positive integers are added in the usual way—1 + 2 + 3 + 4 + …—the partial sums keep increasing without a finite limit. The ordinary sum is therefore not −1/12. That familiar value comes from analytic continuation of the zeta function, a form of regularization with a different meaning from adding each term in the usual way.
What Is the Ordinary Sum of 1 + 2 + 3 + …?
Stop after the positive integer n, and the partial sum is:
Sₙ = 1 + 2 + … + n = n(n + 1)/2
As n increases, Sₙ grows without bound. For example, stopping at 10 gives 55, while stopping at 100 gives 5,050. The sequence of partial sums does not approach a real number; it continues to grow. Under the standard definition of an infinite series, this series diverges and has no finite ordinary sum.
Saying that the sum “goes to infinity” describes the increasing partial sums. It does not mean that infinity is another real number that can be substituted as the sum.
Where Does −1/12 Come From?
The Riemann zeta function begins with the series ζ(s) = 1 + 1/2ˢ + 1/3ˢ + …, which defines the function by this series only when the real part of s is greater than 1. Outside that region, analytic continuation extends the same function, which has a simple pole at s = 1.
Using that continuation at s = −1 gives ζ(−1) = −1/12. This is a value of the continued zeta function, not the limit of the partial sums 1 + 2 + … + n, which still grow without bound. The NIST Digital Library of Mathematical Functions treats the series’ defining region and its values at negative integers separately.
Why Do Popular Derivations Look Like Proofs?
Short explanations often begin with an alternating series such as 1 − 1 + 1 − 1 + …, then shift or group terms to assign a “value” to another series. These steps can illustrate the puzzle, but operations such as adding, subtracting, shifting, or grouping terms in an infinite series are valid only when convergence conditions or the chosen summation method allow them. Rules for finite sums do not prove an ordinary sum for a divergent series.
It helps to separate two questions: “What value do the partial sums approach as terms are added?” and “What value does a specified regularization method assign to this series?” The answers need not be the same.
Does Cesàro Summation Give −1/12?
For first-order Cesàro summation, or (C,1), the method averages partial sums rather than grouping terms directly. Here Sₖ = k(k + 1)/2, so the average from S₀ through Sₙ is σₙ = (S₀ + S₁ + … + Sₙ)/(n + 1) = n(n + 2)/6. This still grows with n and has no finite limit. Under the standard (C,1) definition, the series is therefore not assigned −1/12.
A short description may call −1/12 a Cesàro sum, but that wording does not match the standard (C,1) definition in the NIST DLMF. The distinction here is between the stated mathematical definitions; it does not establish exactly what was said in the video.
Ramanujan Summation and Zeta Regularization
The video title uses “Ramanujan Summation,” while explanations of −1/12 often discuss the zeta function. This article follows the zeta-regularization route because its definitions and value can be checked against direct references. It does not claim that analytic continuation is the only definition of Ramanujan summation, or that the ordinary sum becomes −1/12.
When you see 1 + 2 + 3 + … = −1/12, read it as shorthand that requires a stated summation method. Without that qualification, the equals sign can falsely suggest that the partial sums of positive integers approach a negative number.
Frequently Asked Questions
Is 1 + 2 + 3 + … = −1/12 true or false?
It is false if “sum” means the ordinary infinite series, because its partial sums grow without bound. It is true that ζ(−1) = −1/12 for the analytically continued Riemann zeta function.
Is −1/12 a made-up value?
No. The analytically continued Riemann zeta function has this value at s = −1. Confusion arises when a function value is written as if it were an ordinary sum without stating the method.
Where can I learn more?
Read the NIST DLMF definitions and properties of the Riemann zeta function in the references below. You can also browse the YouTube channel directory or the playlist directory. The linked video and playlist are included with the source information below.
References
- NIST DLMF §25.2: Definition of the Riemann zeta function and analytic continuation
- NIST DLMF §25.6: Values of the zeta function at negative integers
- NIST DLMF §1.15: Definition of Cesàro summation
Scope: The discussion of the ordinary sum, analytic continuation, ζ(−1), and Cesàro summation follows the cited NIST DLMF sections. The video is linked as the topic’s source; this article does not claim to verify its audio.