A363906 Decimal expansion of Sum_{n>=1} (arcsin(1/n) - sin(1/n)).

7, 9, 9, 5, 8, 8, 6, 2, 3, 5, 5, 3, 3, 7, 6, 9, 9, 0, 1, 1, 3, 9, 9, 1, 1, 1, 3, 5, 2, 7, 2, 3, 9, 8, 2, 5, 0, 4, 0, 1, 7, 2, 2, 8, 4, 1, 9, 0, 7, 7, 7, 9, 6, 8, 3, 6, 4, 1, 1, 6, 5, 9, 2, 8, 4, 3, 6, 7, 7, 3, 0, 4, 0, 6, 7, 7, 5, 5, 7, 2, 1, 7, 9, 1, 8, 1, 7

Offset: 0

Author

Bernard Schott, Jun 27 2023

Keywords

• nonn
• cons

Comments

Series Sum_{n>=1} arcsin(1/n) and Sum_{n>=1} sin(1/n) -> oo but with v(n) = (arcsin(1/n) - sin(1/n)), as v(n) ~ 1 / (3*n^3) when n -> oo, the series Sum_{n>=1} v(n) is convergent.

Examples

			0.79958862355337699...
		

Crossrefs

Cf. A096444, A248946, A342680, A362662.

Programs

Mathematica

NSum[ArcSin[1/n]-Sin[1/n], {n, Infinity}, WorkingPrecision -> 95, NSumTerms -> 82] // RealDigits[#, 10, 87] &//First (* Stefano Spezia, Jun 27 2023 *)

PARI

sumpos(n=1, asin(1/n) - sin(1/n)) \\ Michel Marcus, Jun 27 2023

Formula

Equals Sum_{k>=1} (binomial(2*k,k)/((2*k+1)*2^(2*k)) - (-1)^k/(2*k+1)!) * zeta(2*k+1). - Vaclav Kotesovec, Jun 27 2023

Extensions

More terms from Stefano Spezia, Jun 27 2023