<P> contradicting stability . </P> <P> The methods used above to sum 1 + 2 + 3 + ⋯ are either not stable or not linear . </P> <P> 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 nth 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 . </P> <P> 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 . </P>

What is the sum of the series 1+2+3+4+....+20