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

Author

Bernard Schott, Apr 23 2021

Keywords

• nonn
• cons
• more

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.

Examples

			-0.140859...
		

References

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

Links

• 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.

Crossrefs

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

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

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).

Extensions

More terms from Amiram Eldar, Apr 23 2021

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