

A338670


Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).


0




OFFSET

0,2


COMMENTS

The equation of the sinc function is y = sin(x)/x.
Equivalently, sum of f(x) = sinc(x) where x > 0 and f'(x) = 0.  David A. Corneth, May 01 2021
These extreme values are obtained when x_k > 0 is a solution to tan(x) = x (see Chronomath link), or equivalently to y = tanc(x) = tan(x)/x = 1. The corresponding kth extreme value is y_k = sin(x_k)/x_k.
Every extremum y_k = (1)^k/(k*Pi) + O(1/k^2), hence the series Sum_{k > 0} sin(x_k)/x_k is convergent.
However, this series is not absolutely convergent, just as (C_1)/2 diverges where C_1 is the corresponding du BoisReymond constant.


REFERENCES

JeanMarie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.18, pp. 285 and 303.


LINKS



FORMULA

Equals Sum_{k >= 1} sinc(x_k) or Sum_{k >= 1} (1)^k / sqrt(1+(x_k)^2), where x_k is the kth positive root of x = tan(x).


EXAMPLE

0.140859...


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STATUS

approved



