A160755 Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}].

1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49

Offset: 1

Author

Marvin Ray Burns, May 25 2009

Keywords

• nonn
• base
• less

Comments

Adding the series -1+sqrt(2)-3^(1/3)+4^(1/4)..., according to this sequence, 10 billion terms must be added to arrive at 11 accurate digits of the MRB constant.

Examples

			For n=1, a(n)=1 because after 10^1 partial sums of -1+sqrt(2)-3^(1/3)+4^(1/4)... you get one accurate digit of the MRB constant.
For n=2, a(n)=2 because after 10^2 partial sums you get two accurate digits and so on.
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 450. ISBN 0521818052.

Links

• Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Convergence Acceleration of Alternating Series, Experimental Mathematics, 9:1 (2000).
• Eric Weisstein's World of Mathematics, MRB Constant.
• Wikipedia, Mathematical constant

Crossrefs

Cf. A037077 (the MRB constant).

Programs

Mathematica

m = NSum[(-1)^n*(n^(1/n) - 1), {n, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 1000]; Table[-Floor[Log[10, Abs[m - NSum[(-1)^n*(n^(1/n) - 1), {n, 10^a}, Method ->"AlternatingSigns", WorkingPrecision -> 1000]]]], {a,1, 50}]

Extensions

Corrections from Marvin Ray Burns, Jun 05 2009

Link to Wikipedia replaced by up-to-date version; keyword:less added R. J. Mathar, Aug 04 2010

Corrections by Marvin Ray Burns, Aug 21 2010, Jul 15 2012