However, though it is divergent, it can be manipulated to yield a number of mathematically interesting results.
For example, many summation methods are used in mathematics to assign numerical values even to a divergent series.
One obvious method to find the sum of the series would be to treat it like a telescoping series and perform the subtractions in place: On the other hand, a similar bracketing procedure leads to the apparently contradictory result Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value".
This is closely akin to the general problem of conditional convergence, and variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.
By taking the average of these two "values", one can justify that the series converges to 1/2.
Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions: In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century.
After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigour, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.
In the terms of complex analysis, 1/2 is thus seen to be the value at z = −1 of the analytic continuation of the series
The sequence of partial sums of Grandi's series is 1, 0, 1, 0, ..., which clearly does not approach any number (although it does have two accumulation points at 0 and 1).
[4] Further, the terms of Grandi's series can be rearranged to have its accumulation points at any interval of two or more consecutive integer numbers, not only 0 or 1.
Around 1987, Anna Sierpińska introduced Grandi's series to a group of 17-year-old precalculus students at a Warsaw lyceum.
She focused on humanities students with the expectation that their mathematical experience would be less significant than that of their peers studying mathematics and physics, so the epistemological obstacles they exhibit would be more representative of the obstacles that may still be present in lyceum students.
Sierpińska initially expected the students to balk at assigning a value to Grandi's series, at which point she could shock them by claiming that 1 − 1 + 1 − 1 + ··· = 1/2 as a result of the geometric series formula.
Ideally, by searching for the error in reasoning and by investigating the formula for various common ratios, the students would "notice that there are two kinds of series and an implicit conception of convergence will be born".
Sierpińska remarks that a priori, the students' reaction shouldn't be too surprising given that Leibniz and Grandi thought 1/2 to be a plausible result; The students were ultimately not immune to the question of convergence; Sierpińska succeeded in engaging them in the issue by linking it to decimal expansions the following day.
As soon as 0.999... = 1 caught the students by surprise, the rest of her material "went past their ears".
[5] In another study conducted in Treviso, Italy around the year 2000, third-year and fourth-year Liceo Scientifico pupils (between 16 and 18 years old) were given cards asking the following: The students had been introduced to the idea of an infinite set, but they had no prior experience with infinite series.
The 88 responses were categorized as follows: The researcher, Giorgio Bagni, interviewed several of the students to determine their reasoning.
Some 16 of them justified an answer of 0 using logic similar to that of Grandi and Riccati.
Bagni notes that their reasoning, while similar to Leibniz's, lacks the probabilistic basis that was so important to 18th-century mathematics.
[6] Joel Lehmann describes the process of distinguishing between different sum concepts as building a bridge over a conceptual crevasse: the confusion over divergence that dogged 18th-century mathematics.
As a result, many students develop an attitude similar to Euler's: Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Jean-Charles Callet.
Euler had viewed the sum as the evaluation at x = 1 of the geometric series
Lehman argues that seeing such a conflicting outcome in intuitive evaluations may motivate the need for rigorous definitions and attention to detail.