is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2.
One obvious method to attack the series
- 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ...
is to treat it like a telescoping series and perform the subtractions in place:
- (1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0.
On the other hand, a similar bracketing procedure leads to the apparently contradictory result
- 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1.
Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)
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:
- S = 1 − 1 + 1 − 1 + ..., so
- 1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S
- 1 − S = S
- 1 = 2S,
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 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 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,
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 1, 0, 1, 0, ..., which clearly does not approach any number (although it does have two accumulation points at 0 and 1). 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
(in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms) 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.
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." However, the students showed no shock at being told that 1 − 1 + 1 − 1 + · · · = 1⁄2 or even that 1 + 2 + 4 + 8 + · · · = −1. 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;
- "A posteriori, however, the explanation of this lack of shock on the part of the students may be somewhat different. They accepted calmly the absurdity because, after all, 'mathematics is completely abstract and far from reality', and 'with those mathematical transformations you can prove all kinds of nonsense', as one of the boys later said."
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".
- "In 1703, the mathematician Guido Grandi studied the addition: 1 – 1 + 1 – 1 + ... (addends, infinitely many, are always +1 and –1). What is your opinion about it?"
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:
- (26) the result is 0
- (18) the result can be either 0 or 1
- (5) the result does not exist
- (4) the result is 1⁄2
- (3) the result is 1
- (2) the result is infinite
- (30) no answer
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.
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.
- "Since series are generally presented without history and separate from applications, the student must wonder not only "What are these things?" but also "Why are we doing this?" The preoccupation with determining convergence but not the sum makes the whole process seem artificial and pointless to many students—and instructors as well."
As a result, many students develop an attitude similar to Euler's:
- "…problems that arise naturally (i.e., from nature) do have solutions, so the assumption that things will work out eventually is justified experimentally without the need for existence sorts of proof. Assume everything is okay, and if the arrived-at solution works, you were probably right, or at least right enough. …so why bother with the details that only show up in homework problems?"
Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Callet.
Find the value of S if S = 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + .... (up to infinity)
- Bagni, Giorgio T. (2005-06-30). "Infinite Series from History to Mathematics Education" (PDF). International Journal for Mathematics Teaching and Learning. Archived from the original (PDF) on 2006-12-29.
- Davis, Harry F. (May 1989). Fourier Series and Orthogonal Functions. Dover. ISBN 978-0-486-65973-2.
- Devlin, Keith (1994). Mathematics, the science of patterns: the search for order in life, mind, and the universe. Scientific American Library. ISBN 978-0-7167-6022-1.
- Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. Bibcode:1975MathM..48...12G. CiteSeerX 10.1.1.639.6923. doi:10.2307/2690371. JSTOR 2690371.
- Knopp, Konrad (1990) . Theory and Application of Infinite Series. Dover. ISBN 978-0-486-66165-0.
- Hobson, E. W. (1907). The theory of functions of a real variable and the theory of Fourier's series. The University of Michigan Historical Mathematics Collection: Cambridge University Press. section 331.
- Lehmann, Joel (1995). "Converging Concepts of Series: Learning from History". Learn from the Masters. ISBN 0-88385-703-0.
- Sierpińska, Anna (November 1987). "Humanities students and epistemological obstacles related to limits". Educational Studies in Mathematics. 18 (4): 371–396. doi:10.1007/BF00240986. JSTOR 3482354.
- Whittaker, E. T.; Watson, G. N. (1962). A course of modern analysis (4th, reprinted ed.). Cambridge University Press.
- One minus one plus one minus one – Numberphile, Grandi's series