# 1 + 2 + 3 + 4 + · · ·

The sum of all

natural number s**1 + 2 + 3 + 4 + · · ·**, also written:$sum\_\{n=1\}^\{infin\}\; n^1,$

is a

divergent series ; the sum of the first "n" terms in the series can be found using the formula $frac\{n(n+1)\}\{2\}$.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 andstring theory .**Proof of partial sum formula**The proof that the sum of natural numbers up to

**n**is $frac\{n(n+1)\}\{2\}$ can be proven in a number of ways. First, let:$S\_n\; =\; 1\; +\; 2\; +\; 3\; +\; 4\; +\; cdots\; +\; (n-2)\; +\; (n-1)\; +\; n.,$

One can rearrage the terms and write them backwards:

:$S\_n\; =\; n\; +\; (n-1)\; +\; (n-2)\; +\; cdots\; +\; 4\; +\; 3\; +\; 2\; +\; 1.,$

If we add these two, term by term, we arrive at:

:$2S\_n\; =\; underbrace\{(n+1)\; +\; ((n-1)+2)+((n-2)+3)+cdots+(3+(n-2))+(2+(n-1))\; +\; (1+n)\}\_\{n\}$

:$2S\_n\; =\; underbrace\{(n+1)\; +\; (n+1)+(n+1)+cdots+(n+1)+(n+1)\; +\; (n+1)\}\_\{n\}$

:$2S\_n\; =\; ncdot(n\; +\; 1)$

:$S\_n\; =\; frac\{n(n+1)\}\{2\}$

**ummation and analytic continuation of the zeta function**The Ramanujan sum of 1 + 2 + 3 + 4 + · · · is −

^{1}⁄_{12}. [*Hardy p.333*]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 theanalytic continuation of ζ(s) gives ζ(−1) as −^{1}⁄_{12}.**Physics**In

Bosonic string theory we wish 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 $D$ independentquantum harmonic oscillator s, 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 $nhbaromega/2$. So using the divergent series we find that the sum over all harmonics is $-hbaromega\; D/24$. Ultimately it is this fact, combined with theno-ghost 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 .**History**In

Srinivasa Ramanujan 's second letter toG. H. Hardy , dated 27 February 1913::"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. …" [*Berndt et al p.53. The link of "Bromwich" to*]Thomas Bromwich was added and some formatting changes were made.**ee also***

Infinite arithmetic series

*1 − 2 + 3 − 4 + · · ·

*1 + 1 + 1 + 1 + · · ·

*Triangular number **Notes****References**

*cite book |author=Berndt, Bruce C., Srinivasa Ramanujan Aiyangar, and Robert A. Rankin |title=Ramanujan: letters and commentary |year=1995 |publisher=American Mathematical Society |id=ISBN 0-8218-0287-9

*cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id=LCC|QA295|.H29|1967**Further reading***cite journal |last=Lepowsky |first=James |title=Vertex operator algebras and the zeta function |journal=Contemporary Mathematics |volume=248 |year=1999 |pages=327–340 |url=http://arxiv.org/abs/math/9909178

*cite book |last=Zee |first=A. |title=Quantum field theory in a nutshell |year=2003 |publisher=Princeton UP |id=ISBN 0-691-01019-6 See pp. 65–6 on the Casimir effect.

*cite book |last=Zwiebach |first=Barton |title=A First Course in String Theory |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-83143-1 See p. 293.**External links*** [

*http://math.ucr.edu/home/baez/week124.html This Week's Finds in Mathematical Physics (Week 124)*] , [*http://math.ucr.edu/home/baez/week126.html (Week 126)*] , [*http://math.ucr.edu/home/baez/week147.html (Week 147)*]

** [*http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf Euler’s Proof That 1 + 2 + 3 + · · · = −*]^{1}⁄_{12}

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