1 + 2 + 3 + 4 + ⋯

The sum of all natural numbers 1 + 2 + 3 + 4 + · · · is a divergent series. The nth partial sum of the series is the triangular number

$\sum_{k=1}^n k = \frac{n(n+1)}{2},$

which increases without bound as n goes to infinity.

Although the full series may seem at first sight not to have any meaningful value, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory and string theory.

Summability

Unlike its alternating counterpart 1 − 2 + 3 − 4 + · · ·, the series 1 + 2 + 3 + 4 + · · · is not Abel summable. Its generating function

$1+2x+3x^2+4x^3+\cdots = \frac{1}{(1-x)^2}$

has a pole at x = 1.

The series can be summed by zeta function regularization. When the real part of s is greater than 1, the Riemann zeta function of s equals the sum $\sum_{n=1}^\infty {n^{-s}}$. This sum diverges when the real part of s is less than or equal to 1, but when s = −1 then the analytic continuation of ζ(s) gives ζ(−1) as −1/12.

The Ramanujan sum of 1 + 2 + 3 + 4 + · · · is also −1/12. In Srinivasa Ramanujan's second letter to G. H. Hardy, dated 27 February 1913, he wrote:

"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"

Physics

In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particularly the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of $D$ independent quantum harmonic oscillators, where $D$ is the dimension of spacetime. If the fundamental oscillation frequency is $\omega$ then the energy in an oscillator contributing to the $n$th harmonic is $n\hbar\omega/2$. So using the divergent series, the sum over all harmonics is $-\hbar\omega D/24$. Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.

A similar calculation is involved in computing the Casimir force.

See also

• Triangular number
• 1 − 2 + 3 − 4 + · · ·
• 1 − 1 + 2 − 6 + 24 − 120 + · · ·

References

• Berndt, Bruce C., Srinivasa Ramanujan Aiyangar, and Robert A. Rankin (1995). Ramanujan: letters and commentary. American Mathematical Society. ISBN 0-8218-0287-9. 
• Hardy, G.H. (1949). Divergent Series. Clarendon Press. LCC QA295 .H29 1967. 

Further reading

• Lepowsky, James (1999). "Vertex operator algebras and the zeta function". Contemporary Mathematics 248: 327–340. arXiv:math/9909178. 
• Zee, A. (2003). Quantum field theory in a nutshell. Princeton UP. ISBN 0-691-01019-6.  See pp. 65–6 on the Casimir effect.
• Zwiebach, Barton (2004). A First Course in String Theory. Cambridge UP. ISBN 0-521-83143-1.  See p. 293.

External links

• This Week's Finds in Mathematical Physics (Week 124), (Week 126), (Week 147)
  • Euler’s Proof That 1 + 2 + 3 + · · · = −1/12
  • John Baez (September 19, 2008). "My Favorite Numbers: 24".