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In standard mathematics, the series 1+2+3+4+⋯ diverges and has no finite sum.

The Claim

The sum of all natural numbers (1 + 2 + 3 + 4 + ) equals 1/12.

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The Short Version

The claim is not supported as stated. In ordinary mathematics, the series 1+2+3+4+⋯ diverges, so it does not have a finite sum and certainly does not equal -1/12. That number arises only in specialized frameworks such as zeta-function regularization or Ramanujan summation, which are not the same as the usual sum of the series.

Caveats

  • Do not confuse ζ(-1) = -1/12 with the ordinary sum of 1+2+3+⋯; they are different mathematical objects.
  • The claim omits the necessary qualifier that -1/12 is a regularized or analytically continued value under a special convention.
  • Popular presentations often dramatize this result without explaining that the standard partial sums grow without bound.

The Receipts

  1. Riemann Zeta Function

    Wolfram MathWorld

  2. 1 + 2 + 3 + 4 + ⋯

    Wikipedia

  3. How (not) to use zeta function regularization

    arXiv

  4. Riemann zeta function

    Wikipedia

  5. Particular values of the Riemann zeta function

    Wikipedia

  6. Can All the Natural Numbers be Summed?

    UCR Math (University of California, Riverside)

  7. Infinity or -1/12?

    Plus Magazine (University of Cambridge)

  8. Ramanujan summation

    Wikipedia

  9. Zeta Function Regularization

    nLab

  10. Values of the Riemann zeta function at integers

    MATEMÀTIQUES: Universitat Autònoma de Barcelona (journal PDF)

+ 15 more sources — see the full list on Lenz

Filed Under

Natural NumbersRamanujan SummationRiemann Zeta FunctionZeta Function Regularization

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