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

%I A233383 #18 Mar 23 2024 17:32:07
%S A233383 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,
%T A233383 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,
%U A233383 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
%N A233383 Decimal expansion of the absolute value of Sum_{n>=1} (-1)^n*sin(1/n).
%C A233383 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...
%H A233383 oldrinb, <a href="http://math.stackexchange.com/questions/473885/evaluating-the-infinite-series-sum-limits-n-1-infty-sin-frac12n-sin-fra">Evaluating the infinite series sum_n=1^infty [sin(1/2n)-sin(1/(2n+1))]</a>, math.stackexchange, Aug 22 2013
%e A233383 0.550796848133929475510066957...
%p A233383 M := 141 :
%p A233383 Digits := 120 :
%p A233383 s := sin(1/2/n)-sin(1/(2*n+1)) :
%p A233383 add(subs(n=i,s),i=1..M) :
%p A233383 pre := evalf(%) :
%p A233383 zetaM := proc(s,M)
%p A233383     local n ;
%p A233383     Zeta(s)-add(1/n^s,n=1..M) ;
%p A233383     evalf(%) ;
%p A233383 end proc:
%p A233383 for dd from 75 to 90 by 5 do
%p A233383     subs(n=1/eps,s) ;
%p A233383     taylor(%,eps=0,dd+1) ;
%p A233383     t := gfun[seriestolist](%,'ogf') ;
%p A233383     add( op(j,t)*zetaM(j-1,M),j=3..nops(t)) ;
%p A233383     x := pre+% ;
%p A233383     print(x) ;
%p A233383 end do:
%p A233383 # now sum_{n>=1} (-1)^n*sin(1/n) = -0.5570986.
%p A233383 x-sin(1.0) ;
%t A233383 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 *)
%K A233383 cons,nonn
%O A233383 0,1
%A A233383 _R. J. Mathar_, Dec 08 2013