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

1,2

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.

References

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

Links

Table of n, a(n) for n=1..50.

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

Example

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.

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}]

Crossrefs

Cf. A037077 (the MRB constant).

Sequence in context: A228881 A252373 A245338 * A017873 A274016 A291572

Adjacent sequences: A160752 A160753 A160754 * A160756 A160757 A160758

Keyword

nonn,base,less

Author

Marvin Ray Burns, May 25 2009

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

Status

approved