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Analytic Summability Theory

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Summability Calculus


The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose prototypical examples include the Abel summation method, the Cesaro means, and the Borel summability method. As will be established in subsequent chapters, the theory of summability of divergent series is intimately connected to the theory of fractional finite sums. In this chapter, we introduce a generalized definition of series as well as a new summability method for computing the value of series according to such a definition. We show that the proposed summability method is both regular and linear, and that it arises quite naturally in the study of local polynomial approximations of analytic functions. The materials presented in this chapter will be foundational to all subsequent chapters.

Errors like straws, upon the surface flow. He who would search for pearls, must dive below…

John Dryden (1631–1700)

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  1. 1.

    A summability method is called:

    1. 1.

      Regular: If it agrees with ordinary summation whenever a series converges in the classical sense of the word.

    2. 2.

      Linear: If it agrees with the identity \(\sum _{k=0}^\infty g(k)\,+\lambda h(k) = \sum _{k=0}^\infty g(k) +\lambda \,\sum _{k=0}^\infty \,h(k)\).

    3. 3.

      Stable: If it agrees with the identity \(\sum _{k=0}^\infty g(k) = g(0) +\sum _{k=0}^\infty g(k+1)\).

  2. 2.

    The reader may consult the Glossary section at the back of this book for brief definitions of these summability methods.

  3. 3.

    The Nörlund mean is a method of assigning limits to infinite sequences. Here, suppose p j is a sequence of positive terms that satisfies \(\frac {p_n}{\sum _{k=0}^n p_k}\to 0\). Then, the Nörlund mean of a sequence (s 0, s 1, …) is given by \(\lim _{n\to \infty } \frac {p_n s_0 + p_{n-1} s_1+\dotsm +p_0 s_n}{\sum _{k=0}^n p_k}\). The limit of an infinite sequence (s 0, s 1, …) is defined by its Nörlund mean. Therefore, the Nörlund mean interprets the limit of s n as n → using an averaging method.

  4. 4.

    Consult the classic book “Divergent Series” by Hardy [Har49] for a definition of this term.

  5. 5.

    In fact, because A n,k  ≥ 0, it is also totally regular (see [Har49] for a definition of this term).

  6. 6.

    In MATLAB, the function is given by the command: 1/x*psi(1,1/x+1).


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Alabdulmohsin, I.M. (2018). Analytic Summability Theory. In: Summability Calculus. Springer, Cham.

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