cp's OEIS Frontend

A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).

%I A338670 #45 Feb 16 2025 08:34:00
%S A338670 1,4,0,8,5,9
%N A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).
%C A338670 The equation of the sinc function is y = sin(x)/x.
%C A338670 Equivalently, sum of f(x) = sinc(x) where x > 0 and f'(x) = 0. - _David A. Corneth_, May 01 2021
%C A338670 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 k-th extreme value is y_k = sin(x_k)/x_k.
%C A338670 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.
%C A338670 However, this series is not absolutely convergent, just as (C_1)/2 diverges where C_1 is the corresponding du Bois-Reymond constant.
%D A338670 Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.18, pp. 285 and 303.
%H A338670 Serge Mehl, <a href="http://serge.mehl.free.fr/anx/sinxsurx.html">Comportement en zéro de sin(x)/x</a>, ChronoMath.
%H A338670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/duBois-ReymondConstants.html">du Bois-Reymond constants</a>.
%H A338670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SincFunction.html">Sinc Function</a>.
%H A338670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TancFunction.html">Tanc Function</a>.
%F A338670 Equals Sum_{k >= 1} sinc(x_k) or Sum_{k >= 1} (-1)^k / sqrt(1+(x_k)^2), where x_k is the k-th positive root of x = tan(x).
%e A338670 -0.140859...
%Y A338670 Coordinates of the 1st extremum: A115365 (x_1), A213053 (y_1).
%Y A338670 Cf. A062546, A224196, A207528, A243108, A245333.
%K A338670 nonn,cons,more
%O A338670 0,2
%A A338670 _Bernard Schott_, Apr 23 2021
%E A338670 More terms from _Amiram Eldar_, Apr 23 2021
%E A338670 Name clarified by _N. J. A. Sloane_, May 01 2021