This article lists occurrences of the paradoxical infinite "sum" , sometimes called Grandi's series.
Guido Grandi illustrated the series with a parable involving two brothers who share a gem.
Thomson's lamp is a supertask in which a hypothetical lamp is turned on and off infinitely many times in a finite time span. One can think of turning the lamp on as adding 1 to its state, and turning it off as subtracting 1. Instead of asking the sum of the series, one asks the final state of the lamp.
One of the best-known classic parables to which infinite series have been applied, Achilles and the tortoise, can also be adapted to the case of Grandi's series.
The Cauchy product of Grandi's series with itself is .
Several series resulting from the introduction of zeros into Grandi's series have interesting properties; for these see Summation of Grandi's series#Dilution.
Grandi's series is just one example of a divergent geometric series.
The rearranged series occurs in Euler's 1775 treatment of the pentagonal number theorem as the value of the Euler function at q = 1.
The power series most famously associated with Grandi's series is its ordinary generating function,
In his 1822 Théorie Analytique de la Chaleur, Joseph Fourier obtains what is currently called a Fourier sine series for a scaled version of the hyperbolic sine function,
He finds that the general coefficient of sin nx in the series is
For n > 1 the above series converges, while the coefficient of sin x appears as