Cesàro summation
Cesàro summation is a summation method in mathematical analysis that assigns a finite value to some divergent infinite series by computing the limit of the arithmetic means of their partial sums, if the limit exists.[1] For a series \sum_{k=0}^\infty a_k with partial sums S_n = \sum_{k=0}^n a_k, the Cesàro sum is defined as \lim_{n \to \infty} \frac{1}{n+1} \sum_{k=0}^n S_k.[2] Introduced by Italian mathematician Ernesto Cesàro (1859–1906) in his 1890 paper "Sur la multiplication des séries," this method formalizes earlier informal approaches to divergent series dating back to Euler in the 18th century.[3] A classic example is the Grandi series $1 - 1 + 1 - 1 + \cdots, whose partial sums alternate between 1 and 0 and thus diverge, but whose Cesàro means converge to \frac{1}{2}.[1] Cesàro summation is the special case (C, 1) of the more general (C, \alpha) methods, where higher-order means are taken iteratively, and it forms part of the broader theory of regular matrix summation methods that satisfy conditions like those in Toeplitz's theorem for preserving convergent sums.[2] If a series converges in the usual sense to a value s, then its Cesàro sum is also s, making the method consistent with ordinary convergence.[4] The method gained prominence through its applications in Fourier analysis, particularly in ensuring the summability of Fourier series via Cesàro means, as established by Fejér's theorem in 1900, which guarantees pointwise convergence to the function at points of continuity.[1] Cesàro summation has since influenced developments in summability theory, including extensions to matrix series and random fields, and remains a foundational tool for handling oscillatory or slowly converging sequences in analysis and beyond.[3]Background and Motivation
Historical Development
The study of divergent series dates back to the 18th century, when Leonhard Euler explored formal manipulations of such series despite their lack of convergence in the modern sense. In his 1760 paper De seriebus divergentibus, Euler assigned values to divergent series using methods like differencing and acceleration, such as interpreting the Grandi's series $1 - 1 + 1 - 1 + \cdots as equaling \frac{1}{2}, which anticipated later summation techniques by providing a heuristic framework for handling divergence.[5] In the late 19th century, Henri Poincaré advanced the theoretical foundation for divergent series through his work on asymptotic expansions. In his 1886 paper "Sur les intégrales irrégulières" published in Acta Mathematica, Poincaré introduced a rigorous definition of asymptotic series, where a series S_n satisfies x^n (J - S_n) \to 0 as x \to \infty, allowing for term-by-term operations under certain conditions and applying this to approximate solutions of differential equations in celestial mechanics.[6] This framework shifted divergent series from mere formal tools to a structured approach for approximations, indirectly influencing subsequent summability methods by emphasizing their utility in physical and mathematical contexts.[6] The modern method of Cesàro summation emerged in 1890 with Ernesto Cesàro's paper "Sur la multiplication des séries," published in the Bulletin des sciences mathématiques et astronomiques. Cesàro proposed averaging the partial sums of a series to assign a finite value to divergent ones, such as summing $1 - 1 + 1 - 1 + \cdots to \frac{1}{2}, thereby providing a consistent generalization of ordinary convergence while extending it to cases where standard limits fail.[7] This innovation, building on earlier informal treatments, marked the first systematic theory for divergent series summation through arithmetic means.[7] In the early 20th century, G. H. Hardy and others expanded Cesàro's ideas into a broader theory of summability. Hardy's 1911 paper with S. Chapman, "A general view of the theory of summable series," published in the Quarterly Journal of Mathematics, analyzed Cesàro methods alongside others like Hölder means, establishing consistency theorems and conditions for when summability implies convergence, thus solidifying their role in analysis.[8] These developments led to generalizations, including higher-order Cesàro means, and integrated the method into mainstream mathematics by the 1920s.[9]Relation to Series Summability
In the theory of infinite series, ordinary convergence is determined by the behavior of the partial sums s_n = \sum_{k=1}^n a_k, where the series \sum_{k=1}^\infty a_k is said to converge if \lim_{n \to \infty} s_n exists and is finite.[10] This standard criterion, rooted in the work of Cauchy and Weierstrass, provides a precise measure for summation but fails for many series of interest in analysis and its applications.[10] A series is divergent if \lim_{n \to \infty} s_n does not exist, which can occur due to oscillatory behavior of the partial sums (where they fluctuate without settling) or unbounded growth (where |s_n| tends to infinity, possibly in an irregular manner).[10] Cesàro summation addresses this limitation by considering the sequence of arithmetic means of the partial sums, allowing the method to assign a finite value in cases where the original limit diverges, particularly when the averages converge despite the oscillations or controlled growth of s_n.[10] This approach can succeed for series whose partial sums are unbounded but whose averages remain bounded and approach a limit, extending the reach of summation beyond classical convergence. One key advantage of Cesàro summation lies in its regularity: if a series converges in the ordinary sense to a sum s, then its Cesàro sum is also s, ensuring consistency with established results.[10] This property prevents the method from altering sums of convergent series while regularizing select divergent ones, thereby broadening applicability without disrupting foundational theorems.[10] Additionally, as a linear method, it preserves algebraic operations such as addition and scalar multiplication for series that are Cesàro summable, maintaining structural integrity in manipulations akin to those for convergent series.[10] Summability methods like Cesàro represent a conceptual extension of convergence, transforming the problem of assigning values to infinite series into one of alternative limiting processes that capture "average" behavior when direct limits fail.[10] By focusing on means rather than raw partial sums, these methods provide a rigorous framework for handling divergence, influencing developments in Fourier analysis, asymptotic expansions, and other areas where non-standard summation is essential.[10]Standard Cesàro Summation
Definition and Arithmetic Means
Cesàro summation of order 1, denoted as (C,1)-summation, provides a method to assign a sum to a divergent series by considering the arithmetic means of its partial sums. Specifically, for an infinite series \sum_{n=1}^\infty a_n, let s_k = \sum_{n=1}^k a_n denote the k-th partial sum. The series is said to be (C,1)-summable to a value s if the limit \lim_{n \to \infty} \sigma_n^{(1)} exists and equals s, where \sigma_n^{(1)} is the arithmetic mean of the first n partial sums.[11] This approach was originally introduced by Ernesto Cesàro in his 1890 work on the multiplication of series, where he proposed averaging partial sums to extend the notion of convergence beyond ordinary limits. The explicit formula for the Cesàro mean of order 1 is given by \sigma_n^{(1)} = \frac{1}{n} \sum_{k=1}^n s_k, with the series \sum a_n being (C,1)-summable to s if \lim_{n \to \infty} \sigma_n^{(1)} = s. The superscript (1) indicates the order of the method, distinguishing it from higher-order generalizations. This mean \sigma_n^{(1)} directly relates to the sequence of partial sums \{s_k\} by smoothing out oscillations or divergences in \{s_k\} through successive averaging, potentially yielding convergence even when \lim_{k \to \infty} s_k does not exist. An equivalent expression for the Cesàro mean in terms of the series terms is \sigma_n^{(1)} = \frac{1}{n} \sum_{j=1}^n (n - j + 1) a_j. This form arises from interchanging the order of summation in \sum_{k=1}^n s_k = \sum_{j=1}^n a_j (n - j + 1).[2]Partial Sums and Notation
The partial sums of a series \sum_{n=1}^\infty a_n are defined as s_n = \sum_{k=1}^n a_k for each positive integer n, representing the finite sum up to the nth term of the sequence \{a_n\}. These partial sums form the foundation for assessing convergence in the classical sense, where the series converges to a sum s if \lim_{n \to \infty} s_n = s. In the context of Cesàro summation, the partial sums are averaged to form the arithmetic means \sigma_n = \frac{1}{n} \sum_{k=1}^n s_k, and the series is said to be Cesàro summable (or (C,1)-summable) to s if \lim_{n \to \infty} \sigma_n = s. This notation (C,1) specifically denotes the first-order Cesàro method, distinguishing it from higher-order generalizations, while the distinction between the series \sum a_n and its putative sum s emphasizes that Cesàro summation extends beyond ordinary convergence. For practical computation, the partial sums satisfy the recursive relation s_n = s_{n-1} + a_n with initial condition s_0 = 0, allowing efficient incremental updates without recomputing the entire sum each time. The Cesàro means can then be computed iteratively using \sigma_n = \frac{n-1}{n} \sigma_{n-1} + \frac{1}{n} s_n, with \sigma_1 = s_1, which follows directly from the definition of \sigma_n and is particularly useful in numerical implementations where memory and time efficiency are concerns, such as in analyzing large sequences or simulating divergent behaviors. This recursive form avoids storing all prior partial sums, reducing complexity to O(1) per step after initializing the running totals. When the partial sums exhibit certain growth patterns, the Cesàro means may still fail to converge, highlighting the method's limitations. For instance, if s_n grows linearly like cn for some constant c > 0, then \sigma_n \sim \frac{cn}{2}, diverging to infinity and indicating the series is not (C,1)-summable. Similarly, logarithmic growth such as s_n \sim \log n, as occurs in the divergent harmonic series \sum 1/n, yields \sigma_n \sim \log n, which also diverges, though more slowly than the partial sums themselves. These edge cases illustrate how Cesàro summation regularizes some oscillatory divergences but cannot always tame unbounded growth.Illustrative Examples
Grandi's Series
Grandi's series is the divergent infinite series \sum_{n=1}^{\infty} (-1)^{n+1} = 1 - 1 + 1 - 1 + \cdots.[3] The partial sums s_n of this series oscillate without converging, alternating between 1 for odd n = 2k+1 (where s_{2k+1} = 1) and 0 for even n = 2k (where s_{2k} = 0).[3] Cesàro summation assigns a value to this series by taking the limit of the arithmetic means of the partial sums, known as the Cesàro means \sigma_n = \frac{1}{n} \sum_{k=1}^n s_k. For Grandi's series, these means converge to \frac{1}{2}, so the (C,1)-sum is \frac{1}{2}.[3] The following table illustrates the first few partial sums s_n and Cesàro means \sigma_n:| n | s_n | \sigma_n |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 0 | 0.5 |
| 3 | 1 | \frac{2}{3} \approx 0.667 |
| 4 | 0 | 0.5 |
| 5 | 1 | 0.6 |
| 6 | 0 | 0.5 |