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Ramanujan Summation

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Ramanujan Summation of Divergent Series

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2185))

Abstract

In the first two sections of this chapter we recall the Euler-MacLaurin formula and use it to define what Ramanujan, in Chapter VI of his second Notebook, calls the “constant” of a series. But, as Hardy has observed, Ramanujan leaves some ambiguity in the definition of this “constant”. Thus in the third section we interpret this constant as the value of a precise solution of a difference equation. Then we can give in Sect. 1.4 a rigorous definition of the Ramanujan summation and its relation to the usual summation for convergent series.

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Authors and Affiliations

  1. Laboratoire J.A.Dieudonné. CNRS, Université de Nice Côte d’ Azur, Nice, France

    Bernard Candelpergher

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  1. Bernard Candelpergher
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Candelpergher, B. (2017). Ramanujan Summation. In: Ramanujan Summation of Divergent Series. Lecture Notes in Mathematics, vol 2185. Springer, Cham. https://doi.org/10.1007/978-3-319-63630-6_1

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