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%I #28 Nov 11 2016 21:07:47
%S 6,3,7,0,9,2,0,8,5,8,9,8,4,9,4,7,4,7,9,1,1,2,5,5,6,0,8,1,7,1,2,8,4,5,
%T 1,5,5,4,4,0,1,8,3,1,4,0,1,5,9,6,0,4,6,7,2,3,8,7,8,0,0,0,6,5,8,2,2,1,5
%N Decimal expansion of the "value" -Sum_{n>=1} (-1)^n / n^(1/n).
%C The sum actually diverges. But by Cohen Villegas Zagier's acceleration methods for alternating series the sum converges to 0.637092...
%C Challenge: find a convergent expression for this constant. [_Joerg Arndt_, Aug 19 2012]
%H Henri Cohen, Fernando Rodriguez Villegas, Don Zagier, <a href="http://www.emis.de/journals/EM/expmath/volumes/9/9.html">Convergence acceleration of alternating series</a>, Experimental Mathematics, vol.9, no.1, pp.3-12, (2000).
%e 0.637092085898494747911255608171284515544018314015960467238780006582215...
%t digits = 69; a[n_] := 1/n^(1/n); a[0] = 0; Clear[f]; f[n_] := f[n] = (d = (3+Sqrt[8])^n; d = (d+1/d)/2; b = 1; c = d; s = 0; For[k = 0, k <= n-1, k++, c = b-c; s = s+c*a[k]; b = (k+n)*(k-n)*b / ((k+1/2)*(k+1))]; s/d) // RealDigits[#, 10, digits] & // First; f[0] ; f[n = 10] ; While[f[n] != f[n-10], n = n+10]; f[n] (* _Jean-François Alcover_, Mar 06 2013 *)
%t digits = 69; a[n_] := 1/n^(1/n); a[0] = 0; Clear[f]; f[n_] := f[n] = (d = (3+Sqrt[8])^n; d = (d+1/d)/2; b = 1; c = d; s = 0; For[k = 0, k <= n-1, k++, c = b-c; s = s+c*a[k]; b = (k+n)*(k-n)*b / ((k+1/2)*(k+1))]; s/d) // RealDigits[#, 10, digits] & // First; f[0] ; f[n = 10] ; While[ f[n] != f[n-10], n = n+10]; f[n] (* _Jean-François Alcover_, Mar 06 2013 *)
%o (PARI)
%o default(realprecision,99);
%o c=-sumalt(n=1, (-1)^n/sqrtn(n,n)) /* 0.6370920858... */
%o v=Vec(Str(c)); /* ["0", ".", "6", "3", "7", ...] */
%o v=vector(#v-1,n,v[n+1]); v[1]=0;
%o v215608=eval(v) /* sequence of digits */
%o /* _Joerg Arndt_, Aug 19 2012 */
%Y Cf. A073009.
%K cons,nonn
%O 0,1
%A _Balarka Sen_, Aug 17 2012
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