Another 1754 publication also criticized Grandi's series on the basis of its collapse to 0.

Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value".

Grandi's series is just one example of a divergent geometric series.

This is an integral version of Grandi's series.

It's easy to see how terms of Grandi's series can be rearranged to arrive at any integer number, not only 0 or 1.

Most of these, especially the simpler ones with historical parallels, sum Grandi's series to .

In the 1710s, Leibniz described Grandi's series in his correspondence with several other mathematicians.

Therefore, Grandi's series is divergent.

For such a relatively late treatment of Grandi's series, it is surprising that Varignon's report does not even mention Leibniz's earlier work.

Nonetheless, there are many summation methods that respect these manipulations and that do assign a "sum" to Grandi's series.

Grandi's series, a certain infinite mathematical series (after Guido Gandi)

When he had first raised the question of Grandi's series to Leibniz, Wolff was inclined toward skepticism along with Marchetti.

The first of his purposes for this paper was to point out the divergence of Grandi's series and expand on Jacob Bernoulli's 1696 treatment.

In fact, this manipulation can be rigorously justified: there are generalized definitions for the sums of series that do assign Grandi's series the value .

Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Callet.

Around 1987, Anna Sierpińska introduced Grandi's series to a group of 17-year-old precalculus students at a Warsaw lyceum.

Treating Grandi's series as a divergent geometric series we may use the same algebraic methods that evaluate convergent geometric series to obtain a third value:

Leibniz described Grandi's series along with the general problem of convergence and divergence in letters to Nicolaus I Bernoulli in 1712 and early 1713.

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.

In a 1715 letter to Jacopo Riccati, Leibniz mentioned the question of Grandi's series and advertised his own solution in the Acta Eruditorum.

To make the time to read the Traité, Varignon had to escape to the countryside for nearly two months, where he wrote on the topic of Grandi's series in relative isolation.

Several series resulting from the introduction of zeros into Grandi's series have interesting properties; for these see Summation of Grandi's series.

Pierre Varignon (1654-1722) treated Grandi's series in his report Précautions à prendre dans l'usage des Suites ou Series infinies résultantes.

The 19th century is remembered as the approximate period of Cauchy's and Abel's largely successful ban on the use of divergent series, but Grandi's series continued to make occasional appearances.

In mathematics Grandi is best known for his work Flores geometrici (1728), studying the rose curve, a curve which has the shape of a petalled flower, and for Grandi's series.