One obvious method to attack the series is 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". Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value: resulting in S =. The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1. The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. Still, to the extent that it is important to be able to bracket seriesat 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 rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.
Relation to the geometric series
For any number in the interval, the sum to infinity of a geometric series can be evaluated via For any, one thus finds and so the limit of series evaluations is However, as mentioned, the series obtained by switching the limits, is divergent. In the terms of complex analysis, is thus seen to be the value at of the Analytic continuation of the series, which is only defined on the complex unit disk,.
Early ideas
Divergence
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is which clearly does not approach any number. Therefore, Grandi's series is divergent. It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is absolutely convergent. Otherwise these operations can alter the result of summation. 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. For instance, the series is a permutation of Grandi's series in which each value in the rearranged series corresponds to a value that is at most four positions away from it in the original series; its accumulation points are 3, 4, and 5.
Education
Cognitive impact
Around 1987, Anna Sierpińska introduced Grandi's series to a group of 17-year-old precalculus students at a Warsawlyceum. 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 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." However, the students showed no shock at being told that or even that. 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".
Preconceptions
In another study conducted in Treviso, Italy around the year 2000, third-year and fourth-year Liceo Scientifico pupils 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. They were given ten minutes without books or calculators. 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. Others justified 1⁄2 as being the average of 0 and 1. Bagni notes that their reasoning, while similar to Leibniz's, lacks the probabilistic basis that was so important to 18th-century mathematics. He concludes that the responses are consistent with a link between historical development and individual development, although the cultural context is different.
Prospects
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 Callet.
Summability
Related problems
The series 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + .... is also divergent, but some methods may be used to sum it to. This is the square of the value most summation methods assign to Grandi's series, which is reasonable as it can be viewed as the Cauchy product of two copies of Grandi's series.
