Functions and Analysis

   

Ramanujan Summation of the Ln(n) Series

Authors: Jesús Sánchez

In this paper it will calculated that the Ramanujan summation of the Ln(n) series is: lim┬█(n→∞)⁡(Ln(1)+Ln(2)+Ln(3)+⋯Ln(n))=Ln(-γ)=Ln(γ)+(2k+1)πi Being γ the Euler-Mascheroni constant 0.577215... The solution is valid for every integer number k (it has infinite solutions). The series are divergent because Ln(n) tends to infinity as n tends to infinity. But, as in other divergent series, a summation value can be associated to it, using different methods (Cesàro, Abel or Ramanujan). If we take the logarithm of the absolute value (this is, we take only the real part of the solution), the value corresponds to the smooth continuation to the y axis of the curve that calculates the partial sums at every point, as we will see in the paper. lim┬█(n→∞)⁡(Ln|1|+Ln|2|+Ln|3|+⋯Ln|n|)=Ln|-γ|=Ln|γ|

Comments: 9 Pages.

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Submission history

[v1] 2019-01-20 04:27:04

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