In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯ 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 1/2. The series, which can also be written as
∑ n = 0 ∞ ( − 1 ) n , {\displaystyle \sum _{n=0}^{\infty }(-1)^{n},}
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
1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … {\displaystyle 1-1+1-1+1-1+1-1+\ldots }
would be to treat it like a telescoping series and perform the subtractions in place:
( 1 − 1 ) + ( 1 − 1 ) + ( 1 − 1 ) + ( 1 − 1 ) + … = 0 + 0 + 0 + 0 + … = 0. {\displaystyle (1-1)+(1-1)+(1-1)+(1-1)+\ldots =0+0+0+0+\ldots =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. {\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+\ldots =1+0+0+0+\ldots =1.}
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. 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 = 2 S , {\displaystyle {\begin{aligned}S&=1-1+1-1+\ldots ,{\text{ so}}\\1-S&=1-(1-1+1-1+\ldots )=1-1+1-1+\ldots =S\\1-S&=S\\1&=2S,\end{aligned}}}
resulting in S = 1 / 2 {\displaystyle S=1/2} . The same conclusion results from calculating − S {\textstyle -S} (from ( − S = ( 1 − S ) − 1 {\textstyle -S=(1-S)-1} ), subtracting the result from S {\displaystyle S} , and solving 2 S = 1 {\displaystyle 2S=1} . 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 1 − 1 + 1 − 1 + ... has no sum. ... but its sum should be 1/2. 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 r {\displaystyle r} in the interval ( − 1 , 1 ) {\displaystyle (-1,1)} , the sum to infinity of a geometric series can be evaluated via
lim N → ∞ ∑ n = 0 N r n = ∑ n = 0 ∞ r n = 1 1 − r . {\displaystyle \lim _{N\to \infty }\sum _{n=0}^{N}r^{n}=\sum _{n=0}^{\infty }r^{n}={\frac {1}{1-r}}.}
For any ε ∈ ( 0 , 2 ) {\displaystyle \varepsilon \in (0,2)} , one thus finds
∑ n = 0 ∞ ( − 1 + ε ) n = 1 1 − ( − 1 + ε ) = 1 2 − ε , {\displaystyle \sum _{n=0}^{\infty }(-1+\varepsilon )^{n}={\frac {1}{1-(-1+\varepsilon )}}={\frac {1}{2-\varepsilon }},}
and so the limit ε → 0 {\displaystyle \varepsilon \to 0} of series evaluations is
lim ε → 0 lim N → ∞ ∑ n = 0 N ( − 1 + ε ) n = 1 2 . {\displaystyle \lim _{\varepsilon \to 0}\lim _{N\to \infty }\sum _{n=0}^{N}(-1+\varepsilon )^{n}={\frac {1}{2}}.}
However, as mentioned, the series obtained by switching the limits,
lim N → ∞ lim ε → 0 ∑ n = 0 N ( − 1 + ε ) n = ∑ n = 0 ∞ ( − 1 ) n {\displaystyle \lim _{N\to \infty }\lim _{\varepsilon \to 0}\sum _{n=0}^{N}(-1+\varepsilon )^{n}=\sum _{n=0}^{\infty }(-1)^{n}}
is divergent. In the terms of complex analysis, 1/2 is thus seen to be the value at z = −1 of the analytic continuation of the power series ∑ n = 0 ∞ z n {\displaystyle \textstyle \sum _{n=0}^{\infty }z^{n}} , which is only defined on the complex unit disk, |z| ≤ 1.
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 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
1 + 1 + 1 + 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 + 1 − ⋯ {\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots }
(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 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".
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:
"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.
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 x = 1 of the geometric series 1 − x + x 2 − x 3 + ⋯ = 1 / ( 1 + x ) {\displaystyle 1-x+x^{2}-x^{3}+\cdots =1/(1+x)} , giving the sum 1/2. However, Callet pointed out that one could instead view Grandi's series as the evaluation at x = 1 of a different series, 1 − x 2 + x 3 − x 5 + x 6 − ⋯ = 1 + x 1 + x + x 2 {\displaystyle 1-x^{2}+x^{3}-x^{5}+x^{6}-\cdots ={\tfrac {1+x}{1+x+x^{2}}}} , giving the sum 2/3. 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 1 − 1 + 1 − 1 + ⋯ being assigned a value of 1⁄2 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 1 − 1 + 1 − 1 + ⋯ 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.
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 n, the average σn of the first n partial sums, and takes the limit of these Cesàro means as n goes to infinity. For Grandi's series, the sequence of arithmetic means is
1, 1⁄2, 2⁄3, 2⁄4, 3⁄5, 3⁄6, 4⁄7, 4⁄8, ... or, more suggestively,
(1⁄2+1⁄2), 1⁄2, (1⁄2+1⁄6), 1⁄2, (1⁄2+1⁄10), 1⁄2, (1⁄2+1⁄14), 1⁄2, ... where
σ n = 1 2 {\displaystyle \sigma _{n}={\frac {1}{2}}} for even n and σ n = 1 2 + 1 2 n {\displaystyle \sigma _{n}={\frac {1}{2}}+{\frac {1}{2n}}} for odd n. This sequence of arithmetic means converges to 1⁄2, so the Cesàro sum of Σak is 1⁄2. Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1, 0, ⋯ is 1⁄2. The Cesàro sum of 1 + 0 − 1 + 1 + 0 − 1 + ⋯ is 2⁄3. So the Cesàro sum of a series can be altered by inserting infinitely many 0s as well as infinitely many brackets. The series can also be summed by the more general fractional (C, a) methods.
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, and in practice he used it almost exclusively 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
A ∑ n = 0 ∞ ( − 1 ) n = lim x → 1 ∑ n = 0 ∞ ( − x ) n = lim x → 1 1 1 + x = 1 2 . {\displaystyle A\sum _{n=0}^{\infty }(-1)^{n}=\lim _{x\rightarrow 1}\sum _{n=0}^{\infty }(-x)^{n}=\lim _{x\rightarrow 1}{\frac {1}{1+x}}={\frac {1}{2}}.}
Related series The corresponding calculation that the Abel sum of 1 + 0 − 1 + 1 + 0 − 1 + ⋯ is 2⁄3 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 1⁄4.
Dilution
Alternating spacing That the ordinary Abel sum of 1 + 0 − 1 + 1 + 0 − 1 + ⋯ is 2⁄3 can also be phrased as the (A, λ) sum of the original series 1 − 1 + 1 − 1 + ⋯ where (λn) = (0, 2, 3, 5, 6, ...). Likewise the (A, λ) sum of 1 − 1 + 1 − 1 + ⋯ where (λn) = (0, 1, 3, 4, 6, ...) is 1⁄3.
Power-law spacing
Exponential spacing The summability of 1 − 1 + 1 − 1 + ⋯ 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:
0 + 1 − 1 + 0 + 1 + 0 + 0 + 0 − 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 + 0 + ⋯. 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 22m−1 ≤ n ≤ 22m − 1 following a (− 1) term, the nth arithmetic means vary over the range
2 3 ( 2 2 m − 1 2 2 m + 2 ) t o 1 3 ( 1 − 2 − 2 m ) , {\displaystyle {\frac {2}{3}}\left({\frac {2^{2m}-1}{2^{2m}+2}}\right)\;\mathrm {to} \;{\frac {1}{3}}(1-2^{-2m}),}
or about 2⁄3 to 1⁄3. In fact, the exponentially spaced series is not Abel summable either. Its Abel sum is the limit as x approaches 1 of the function
F(x) = 0 + x − x2 + 0 + x4 + 0 + 0 + 0 − x8 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + x16 + 0 + ⋯. This function satisfies a functional equation:
F ( x ) = x − x 2 + x 4 − x 8 + ⋯ = x − [ ( x 2 ) − ( x 2 ) 2 + ( x 2 ) 4 − ⋯ ] = x − F ( x 2 ) . {\displaystyle {\begin{array}{rcl}F(x)&=&\displaystyle x-x^{2}+x^{4}-x^{8}+\cdots \\[1em]&=&\displaystyle x-\left[(x^{2})-(x^{2})^{2}+(x^{2})^{4}-\cdots \right]\\[1em]&=&\displaystyle x-F(x^{2}).\end{array}}}
This functional equation implies that F(x) roughly oscillates around 1⁄2 as x approaches 1. To prove that the amplitude of oscillation is nonzero, it helps to separate F into an exactly periodic and an aperiodic part:
F ( x ) = Ψ ( x ) + Φ ( x ) {\displaystyle F(x)=\Psi (x)+\Phi (x)}
where
Φ ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! ( 1 + 2 n ) ( log 1 x ) n {\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!(1+2^{n})}}\left(\log {\frac {1}{x}}\right)^{n}}
satisfies the same functional equation as F. This now implies that Ψ(x) = −Ψ(x2) = Ψ(x4), so Ψ is a periodic function of loglog(1/x). Since dy (p.77) speaks of "another solution" and "plainly not constant", although technically he does not prove that F and Φ are different. Since the Φ part has a limit of 1⁄2, F oscillates as well.
Separation of scales Given any function φ(x) such that φ(0) = 1, and the derivative of φ is integrable over (0, +∞), then the generalized φ-sum of Grandi's series exists and is equal to 1⁄2:
S φ = lim δ ↓ 0 ∑ k = 0 ∞ ( − 1 ) k φ ( δ k ) = 1 2 . {\displaystyle S_{\varphi }=\lim _{\delta \downarrow 0}\sum _{k=0}^{\infty }(-1)^{k}\varphi (\delta k)={\frac {1}{2}}.}
The Cesàro or Abel sum is recovered by letting φ be a triangular or exponential function, respectively. If φ 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:
S φ = lim δ ↓ 0 ∑ k = 0 ∞ [ φ ( 2 k δ ) − φ ( 2 k δ + δ ) ] = lim δ ↓ 0 ∑ k = 0 ∞ φ ′ ( 2 k δ + c k ) ( − δ ) = − 1 2 ∫ 0 ∞ φ ′ ( x ) d x = − 1 2 φ ( x ) | 0 ∞ = 1 2 . {\displaystyle {\begin{array}{rcl}S_{\varphi }&=&\displaystyle \lim _{\delta \downarrow 0}\sum _{k=0}^{\infty }\left[\varphi (2k\delta )-\varphi (2k\delta +\delta )\right]\\[1em]&=&\displaystyle \lim _{\delta \downarrow 0}\sum _{k=0}^{\infty }\varphi '(2k\delta +c_{k})(-\delta )\\[1em]&=&\displaystyle -{\frac {1}{2}}\int _{0}^{\infty }\varphi '(x)\,dx=-{\frac {1}{2}}\varphi (x)|_{0}^{\infty }={\frac {1}{2}}.\end{array}}}
Euler transform and analytic continuation
Borel sum The Borel sum of Grandi's series is again 1⁄2, since
1 − x + x 2 2 ! − x 3 3 ! + x 4 4 ! − ⋯ = e − x {\displaystyle 1-x+{\frac {x^{2}}{2!}}-{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}-\cdots =e^{-x}}
and
∫ 0 ∞ e − x e − x d x = ∫ 0 ∞ e − 2 x d x = 1 2 . {\displaystyle \int _{0}^{\infty }e^{-x}e^{-x}\,dx=\int _{0}^{\infty }e^{-2x}\,dx={\frac {1}{2}}.}
The series can also be summed by generalized (B, r) methods.
Spectral asymmetry The entries in Grandi's series can be paired to the eigenvalues of an infinite-dimensional operator on Hilbert space. Giving the series this interpretation gives rise to the idea of spectral asymmetry, which occurs widely in physics. The value that the series sums to depends on the asymptotic behaviour of the eigenvalues of the operator. Thus, for example, let { ω n } {\displaystyle \{\omega _{n}\}} be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum
∑ n sgn ( ω n ) {\displaystyle \sum _{n}\operatorname {sgn}(\omega _{n})\;}
where sgn ( ω n ) = ± 1 {\displaystyle \operatorname {sgn}(\omega _{n})=\pm 1} is the sign of the eigenvalue. The series can be given concrete values by considering various limits. For example, the heat kernel regulator leads to the sum
lim t → 0 ∑ n sgn ( ω n ) e − t | ω n | {\displaystyle \lim _{t\to 0}\sum _{n}\operatorname {sgn}(\omega _{n})e^{-t|\omega _{n}|}}
which, for many interesting cases, is finite for non-zero t, and converges to a finite value in the limit.
Methods that fail The integral function method with pn = exp (−cn2) and c > 0. The moment constant method with
d χ = e − k ( log x ) 2 x − 1 d x {\displaystyle d\chi =e^{-k(\log x)^{2}}x^{-1}dx}
and k > 0.
Geometric series The geometric series in ( x − 1 ) {\displaystyle (x-1)} ,
1 x = 1 − ( x − 1 ) + ( x − 1 ) 2 − ( x − 1 ) 3 + ( x − 1 ) 4 − . . . {\displaystyle {\frac {1}{x}}=1-(x-1)+(x-1)^{2}-(x-1)^{3}+(x-1)^{4}-...}
is convergent for | x − 1 | < 1 {\displaystyle |x-1|<1} . Formally substituting x = 2 {\displaystyle x=2} would give
1 2 = 1 − 1 + 1 − 1 + 1 − . . . {\displaysty