In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the
infinite series
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
, also written
:
is sometimes called Grandi's series, after Italian
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
,
philosopher
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
, and priest
Guido Grandi, who gave a memorable treatment of the series in 1703. It is a
divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mus ...
, meaning that the sequence of partial sums of the series does not converge.
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. For example, the
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum ...
and the
Ramanujan summation of this series are both .
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
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Definition
More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if
\lim_\,\s ...
, and variations of this idea, called the
Eilenberg–Mazur swindle, are sometimes used in
knot theory
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
and
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. 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
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mat ...
, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s.
[; ]
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
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, is thus seen to be the value at of the
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
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 sum
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
s, 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 point
In mathematics, a limit point, accumulation point, or cluster point of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood of ...
s 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
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely ma ...
) is a
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
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
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at ...
lyceum
The lyceum is a category of educational institution defined within the education system of many countries, mainly in Europe. The definition varies among countries; usually it is a type of secondary school. Basic science and some introduction to ...
. 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
Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowled ...
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;
: "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 caught the students by surprise, the rest of her material "went past their ears".
Preconceptions
In another study conducted in
Treviso
Treviso ( ; ; ) is a city and (municipality) in the Veneto region of northern Italy. It is the capital of the province of Treviso and the municipality has 87.322 inhabitants (as of December 2024). Some 3,000 live within the Venetian wall ...
,
Italy
Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
around the year 2000, third-year and fourth-year ''
Liceo Scientifico'' pupils (between 16 and 18 years old) were given cards asking the following:
: "In 1703, the mathematician Guido Grandi studied the addition: (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
: (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.
: "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 Jean-Charles Callet. Euler had viewed the sum as the evaluation at of the
geometric series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
, 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
Related problems
The series (
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
infinity) 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
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Definitions
The Cauchy product may apply to infin ...
of two copies of Grandi's series.
See also
*
1 − 1 + 2 − 6 + 24 − 120 + ⋯
*
1 + 1 + 1 + 1 + ⋯
*
1 − 2 + 3 − 4 + ⋯
*
1 + 2 + 3 + 4 + ⋯
The infinite series whose terms are the positive integers is a divergent series. The ''n''th partial sum of the series is the triangular number
\sum_^n k = \frac,
which increases without bound as ''n'' goes to infinity. Because the sequence of ...
*
1 + 2 + 4 + 8 + ⋯
*
1 − 2 + 4 − 8 + ⋯
*
Ramanujan summation
*
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum ...
*
Thomson's lamp
Notes
References
*
*
*
*
*
*
*
*
*
*
External links
One minus one plus one minus one – Numberphile Grandi's series
{{Grandi's series
Divergent series
Geometric series
1 (number)
Mathematical paradoxes
Parity (mathematics)