Occurrences of Grandi's series

This article lists occurrences of the paradoxical infinite "sum" 1 − 1 + 1 − 1 + ⋯ , sometimes called Grandi's series.

Guido Grandi illustrated the series with a parable involving two brothers who share a gem.

The ordinary, Cesàro, and Abel sums of this series involve limits of the Dirichlet, Fejér, and Poisson kernels, respectively.

Keeping Grandi's series in mind, this relation explains why ζ(0) = −1⁄2; see also 1 + 1 + 1 + 1 + · · ·.

[9] Infinite-dimensional real projective space RP∞ is another structure with one cell of every dimension and therefore an Euler characteristic of 1 − 1 + 1 − · · ·.

This space can be described as the quotient of the infinite-dimensional sphere by identifying each pair of antipodal points.

[11] Grandi's series, and generalizations thereof, occur frequently in many branches of physics; most typically in the discussions of quantized fermion fields (for example, the chiral bag model), which have both positive and negative eigenvalues; although similar series occur also for bosons, such as in the Casimir effect.

The Grandi series has been applied to e.g. ballet by Benjamin Jarvis, in The Invariant journal.

PDF here: https://invariants.org.uk/assets/TheInvariant_HT2016.pdf The noise artist Jliat has a 2000 musical single Still Life #7: The Grandi Series advertised as "conceptual art"; it consists of nearly an hour of silence.

Binomial coefficients C ( n , k ) extended for negative and fractional n , illustrated with a simple binomial . It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case n = −1 gives Grandi's series .