A342680 Decimal expansion of Sum_{n>=1} sin(sin(n)/n).

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

Offset: 0

Author

Bernard Schott, Mar 18 2021

Keywords

• nonn
• cons

Comments

Abel summation shows the series is convergent.

Examples

			0.96139431594573654724764595316154730686858269301058...
		

References

• Konrad Knopp, Theory and Application of Infinite Series, Blackie, 1928, p. 313.
• Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice C.3.7 2.3.b)4. p. 309.

Links

• Wikipedia, Summation by parts.

Crossrefs

Cf. A070751, A070752, A096418, A248946.

Programs

Magma

nDgtsOutput:=110; nDgtsPrecision:=nDgtsOutput+10; SetDefaultRealField(RealField(nDgtsPrecision)); kMax:=Ceiling(1.395*nDgtsPrecision-3); mMax:=Ceiling(1.5*kMax); sum:=0.0; S1:=[0.0 : j in [1..kMax]]; n:=0; for m in [1..mMax] do S2:=S1; for k in [1..355] do n:=n+1; sum+:=Sin(Sin(n)/n); end for; S1[1]:=sum; for j in [1..kMax-1] do S1[j+1]:=(S2[j]+S1[j])/2; end for; end for; ChangePrecision(S1[#S1], nDgtsOutput); // The constants 1.395 and 1.5 were empirically derived; 355 is used because 355/Pi is very close to an odd integer. - Jon E. Schoenfield, Mar 21 2021

Extensions

a(3)-a(104) from Jon E. Schoenfield, Mar 20 2021