-1/12

The infinite series whose terms are the natural numbers **1 + 2 + 3 + 4 + ⋯** is a divergent series. The *n*th partial sum of the series is the triangular number in figure 1

which increases without bound as *n* goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.

Although the series seems at first sight not to have any meaningful value at all, 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. Many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −
1/12, which is expressed by a famous formula:

figure 2

where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning.

In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".

## Partial sums

The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc. The *n*th partial sum is given by a simple formula:

figure 1

This equation was known to the Pythagoreans as early as the sixth century BCE.Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.

The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test.

## 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 *ω* then the energy in an oscillator contributing to the *n*th harmonic is *nħω*/2. So using the divergent series, the sum over all harmonics is − *ħω*(*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.(infinite fractal fruit of life)

The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension.An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates. The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term − 1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.

A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.