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

1, 4, 0, 8, 5, 9

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 k-th 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 Bois-Reymond constant.

References

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

Links

Table of n, a(n) for n=0..5.

Serge Mehl, Comportement en zéro de sin(x)/x, ChronoMath.

Eric Weisstein's World of Mathematics, du Bois-Reymond constants.

Eric Weisstein's World of Mathematics, Sinc Function.

Eric Weisstein's World of Mathematics, Tanc Function.

Formula

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).

Example

-0.140859...

Crossrefs

Coordinates of the 1st extremum: A115365 (x_1), A213053 (y_1).

Cf. A062546, A224196, A207528, A243108, A245333.

Sequence in context: A245174 A109169 A011291 * A233590 A078889 A176534

Adjacent sequences: A338667 A338668 A338669 * A338671 A338672 A338673

Keyword

nonn,cons,more

Author

Bernard Schott, Apr 23 2021

Extensions

More terms from Amiram Eldar, Apr 23 2021

Name clarified by N. J. A. Sloane, May 01 2021

Status

approved