A233383 Decimal expansion of the absolute value of Sum_{n>=1} (-1)^n*sin(1/n).

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

Offset: 0

Author

R. J. Mathar, Dec 08 2013

Keywords

• cons
• nonn

Comments

If the contribution of the first term, -sin(1) = -A049469, is omitted, the constant becomes Sum_{n>=1} (sin(1/(2n)) - sin(1/(2n+1))) = 0.29067413667396703114243536419518058522...

Examples

			0.550796848133929475510066957...
		

Links

• oldrinb, Evaluating the infinite series sum_n=1^infty [sin(1/2n)-sin(1/(2n+1))], math.stackexchange, Aug 22 2013

Programs

Maple

M := 141 :
Digits := 120 :
s := sin(1/2/n)-sin(1/(2*n+1)) :
add(subs(n=i,s),i=1..M) :
pre := evalf(%) :
zetaM := proc(s,M)
    local n ;
    Zeta(s)-add(1/n^s,n=1..M) ;
    evalf(%) ;
end proc:
for dd from 75 to 90 by 5 do
    subs(n=1/eps,s) ;
    taylor(%,eps=0,dd+1) ;
    t := gfun[seriestolist](%,'ogf') ;
    add( op(j,t)*zetaM(j-1,M),j=3..nops(t)) ;
    x := pre+% ;
    print(x) ;
end do:
# now sum_{n>=1} (-1)^n*sin(1/n) = -0.5570986.
x-sin(1.0) ;

Mathematica

digits = 105; NSum[(-1)^n*Sin[1/n], {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 24 2014 *)