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

Because the series 1 + 2 + 3 + 4 + · · · diverges, it does not have a sum in the usual sense of the word. Nonetheless, certain generalized summation methods for divergent series do assign the series of "sum" of −1/12. Unlike its alternating counterpart 1 − 2 + 3 − 4 + · · ·, the series 1 + 2 + 3 + 4 + · · · is not Abel summable, so more advanced methods are required.

Heuristics

Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = −1/12" in chapter 8 of his first notebook.^[1]^[2]^[3] The simpler, less rigorous derivation proceeds as follows. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯. Then multiply this equation by 4 and subtract:

{\begin{alignedat}{7}c&{}={}&1+2&&{}+3+4&&{}+5+6+\cdots \\4c&{}={}&4&&{}+8&&{}+12+\cdots \\-3c&{}={}&1-2&&{}+3-4&&{}+5-6+\cdots \\\end{alignedat}}

Now the task is to sum the alternating series 1 − 2 + 3 − 4 + · · ·. This turns out to be an easier task, as it resembles the power series expansion of the function 1/(1 + x)² with 1 substituted for x, so:

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

Dividing both sides by −3, one gets c = −1/12.

Zeta function regularization

The series 1 + 2 + 3 + 4 + · · · can be 'summed' by zeta function regularization. When the real part of s is greater than 1, the Riemann zeta function ζ(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, so, in particular, the series 1 + 2 + 3 + 4 + · · · that results from setting s = –1 does not converge in the ordinary sense. It is only by extending ζ using analytic continuation that we find ζ(−1) = −1/12. (More generally, ζ(s) will always be given by the degree zero term of the Laurent series expansion around h = 0 of $\sum _{n=1}^{\infty }{n^{-s}}e^{hn}$ .)

One way to compute ζ(−1) is to use the relationship between the Riemann zeta function and the Dirichlet eta function. Where both Dirichlet series converge, one has the identities:

{\begin{alignedat}{8}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\cdot 2^{-s}\zeta (s)&{}={}&2\cdot 2^{-s}&&{}+2\cdot 4^{-s}&&{}+2\cdot 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s)\\\end{alignedat}}

The identity $(1-2^{1-s})\zeta (s)=\eta (s)$ continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. Substituting s = −1, one gets −3ζ(−1)=η(−1)=1/4 and so ζ(−1) = −1/12.

Ramanujan summation

The Ramanujan sum of 1 + 2 + 3 + 4 + · · · is also −1/12. In 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. …"^[4]

David Leavitt's novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this passage.^[5]

Physics

In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particular the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of $D-2$ independent quantum harmonic oscillators, one for each transverse direction, 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 $-\hbar \omega (D-2)/24$ . 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, using the Epstein zeta-function in place of the Riemann zeta function, is involved in computing the Casimir force.^[6]

Notes

^ Ramanujan's Notebooks, retrieved January 26, 2014
^ Abdi, Wazir Hasan (1992), Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician, National, p. 41
^ Berndt, Bruce C. (1985), Ramanujan’s Notebooks: Part 1, Springer-Verlag, pp. 135–136
^ Berndt et al. p.53.
^ Leavitt, David (2007), The Indian Clerk, Bloomsbury, pp. 61–62
^ Zeidler, Eberhard (2007), Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists, Springer, pp. 305–306, ISBN 9783540347644.

References

Berndt, Bruce C., Srinivasa Ramanujan Aiyangar, and Robert A. Rankin (1995). Ramanujan: letters and commentary. American Mathematical Society. ISBN 0-8218-0287-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
Hardy, G.H. (1949). Divergent Series. Clarendon Press. LCC QA295 .H29 1967.

External links

This Week's Finds in Mathematical Physics (Week 124), (Week 126), (Week 147)
- Euler’s Proof That 1 + 2 + 3 + · · · = −1/12 - By John Baez
- John Baez (September 19, 2008). "My Favorite Numbers: 24" (PDF).
The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation by Terence Tao
A recursive evaluation of zeta of negative integers by Luboš Motl
ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12 Numberphile video with over a million views
- Sum of Natural Numbers (second proof and extra footage) includes demonstration of Euler's method.
- What do we get if we sum all the natural numbers? response to comments about video by Tony Padilla
Divergent Series: why 1 + 2 + 3 + · · · = −1/12 by Brydon Cais from University of Arizona

[1] Ramanujan's Notebooks, retrieved January 26, 2014

[2] Abdi, Wazir Hasan (1992), Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician, National, p. 41

[3] Berndt, Bruce C. (1985), Ramanujan’s Notebooks: Part 1, Springer-Verlag, pp. 135–136

[4] Berndt et al. p.53.

[5] Leavitt, David (2007), The Indian Clerk, Bloomsbury, pp. 61–62

[6] Zeidler, Eberhard (2007), Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists, Springer, pp. 305–306, ISBN 9783540347644.

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