A215608 Decimal expansion of the "value" -Sum_{n>=1} (-1)^n / n^(1/n).

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, 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

Offset

0,1

Comments

The sum actually diverges. But by Cohen Villegas Zagier's acceleration methods for alternating series the sum converges to 0.637092...

Challenge: find a convergent expression for this constant. [Joerg Arndt, Aug 19 2012]

Links

Table of n, a(n) for n=0..68.

Henri Cohen, Fernando Rodriguez Villegas, Don Zagier, Convergence acceleration of alternating series, Experimental Mathematics, vol.9, no.1, pp.3-12, (2000).

Example

0.637092085898494747911255608171284515544018314015960467238780006582215...

Mathematica

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 *)
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 *)

Prog

(PARI)
default(realprecision, 99);
c=-sumalt(n=1, (-1)^n/sqrtn(n, n)) /* 0.6370920858... */
v=Vec(Str(c)); /* ["0", ".", "6", "3", "7", ...] */
v=vector(#v-1, n, v[n+1]);  v[1]=0;
v215608=eval(v) /* sequence of digits */
/* Joerg Arndt, Aug 19 2012 */

Crossrefs

Cf. A073009.

Sequence in context: A024563 A319232 A265179 * A232883 A119643 A187747

Adjacent sequences: A215605 A215606 A215607 * A215609 A215610 A215611

Keyword

cons,nonn

Author

Balarka Sen, Aug 17 2012

Status

approved