In mathematics, the infinite series is a divergent series, meaning that the sequence of partial sums of the series does not converge. Although it is divergent, it can be manipulated to yield a number of mathematically interesting results. In particular, various summation methods (techniques for assigning numerical values even to a divergent series) assign this series the value .
The series, which can also be written as
is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703.
Nonrigorous methods 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 .
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 . The same conclusion results from calculating (from (), subtracting the result from , and solving .
The above manipulations do not consider what the sum of a series rigorously 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: The series has no sum. ... but its sum should be . 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.;
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 power 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 (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 – the infinitude of both +1s and −1s allows any finite number of 1s or −1s to be prepended, by Hilbert's paradox of the Grand Hotel) 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 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 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 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 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 (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. 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 (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 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 Jean-Charles Callet. Euler had viewed the sum as the evaluation at of the geometric series , giving the sum . However, Callet pointed out that one could instead view Grandi's series as the evaluation at of a different series, , giving the sum . Lehman argues that seeing such a conflicting outcome in intuitive evaluations may motivate the need for rigorous definitions and attention to detail.
Summability
General considerations
Stability and linearity The formal manipulations that lead to being assigned a value of include:
Adding or subtracting two series term-by-term, Multiplying through by a scalar term-by-term, "Shifting" the series with no change in the sum, and Increasing the sum by adding a new term to the series' head.
These are all legal manipulations for sums of convergent series, but is not a convergent series.
Nonetheless, there are many summation methods that respect these manipulations and that do assign a "sum" to Grandi's series. Two of the simplest methods are Cesàro summation and Abel summation.Davis pp.152, 153, 157
Cesàro sum The first rigorous method for summing divergent series was published by Ernesto Cesàro in 1890. The basic idea is similar to Leibniz's probabilistic approach: essentially, the Cesàro sum of a series is the average of all of its partial sums. Formally one computes, for each , the average of the first partial sums, and takes the limit of these Cesàro means as goes to infinity.
For Grandi's series, the sequence of arithmetic means is
or, more suggestively,
where
This sequence of arithmetic means converges to , so the Cesàro sum of is . Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1, 0, ⋯ is .Davis pp.153, 163
The Cesàro sum of is . So the Cesàro sum of a series can be altered by inserting infinitely many 0s as well as infinitely many brackets.Davis pp.162-163, ex.1-5
The series can also be summed by the more general fractional (C, a) methods.Smail p.131
Abel sum Abel summation is similar to Euler's attempted definition of sums of divergent series, but it avoids Callet's and N. Bernoulli's objections by precisely constructing the function to use. In fact, Euler likely meant to limit his definition to power series,Kline 1983 p.313 and in practice he used it almost exclusivelyBromwich p.322 in a form now known as Abel's method.
Given a series a0 + a1 + a2 + ⋯, one forms a new series a0 + a1x + a2x2 + ⋯. If the latter series converges for 0 < x < 1 to a function with a limit as x tends to 1, then this limit is called the Abel sum of the original series, after Abel's theorem which guarantees that the procedure is consistent with ordinary summation. For Grandi's series one has
Davis p.159
Related series The corresponding calculation that the Abel sum of is involves the function (1 + x)/(1 + x + x2).
Whenever a series is Cesàro summable, it is also Abel summable and has the same sum. On the other hand, taking the Cauchy product of Grandi's series with itself yields a series which is Abel summable but not Cesàro summable: 1 − 2 + 3 − 4 + ⋯ has Abel sum .Davis p.165
Dilution
Alternating spacing That the ordinary Abel sum of is can also be phrased as the (A, λ) sum of the original series where (λn) = (0, 2, 3, 5, 6, ...). Likewise the (A, λ) sum of where (λn) = (0, 1, 3, 4, 6, ...) is .Hardy p.73
Exponential spacing The summability of can be frustrated by separating its terms with exponentially longer and longer groups of zeros. The simplest example to describe is the series where (−1)n appears in the rank 2n:
This series is not Cesàro summable. After each nonzero term, the partial sums spend enough time lingering at either 0 or 1 to bring the average partial sum halfway to that point from its previous value. Over the interval following a term, the th arithmetic means vary over the range
or about to .Hardy p.60
In fact, the exponentially spaced series is not Abel summable either. Its Abel sum is the limit as x approaches 1 of the function
This function satisfies a functional equation:
This functional equation implies that roughly oscillates around as approaches 1. To prove that the amplitude of oscillation is nonzero, it helps to separate into an exactly periodic and an aperiodic part:
where
satisfies the same functional equation as . This now implies that , so is a periodic function of . Since dy (p. 77) speaks of "another solution" and "plainly not constant", although technically he does not prove that and are different. Since the part has a limit of , oscillates as well.
Separation of scales Given any function such that , such that the derivative of is integrable over , then the generalized -sum of Grandi's series exists and is equal to :
The Cesàro or Abel sum is recovered by letting be a triangular or exponential function, respectively. If is additionally assumed to be continuously differentiable, then the claim can be proved by applying the mean value theorem and converting the sum into an integral. Briefly:
Saichev pp.260-262
Borel sum The Borel sum of Grandi's series is again , since
and