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A173273 Decimal expansion of  M2 = 1 - 2*M where M is the MRB constant A037077. 1
6, 2, 4, 2, 8, 0, 7, 1, 5, 0, 7, 5, 8, 6, 5, 7, 5, 9, 5, 0, 2, 9, 6, 4, 1, 3, 1, 8, 9, 1, 4, 5, 3, 5, 3, 9, 8, 8, 8, 1, 9, 3, 8, 1, 0, 1, 9, 9, 7, 2, 2, 4, 2, 7, 6, 5, 5, 9, 9, 0, 6, 3, 1, 8, 2, 1, 0, 4, 5, 5, 3, 6, 8, 7, 0, 6, 7, 9, 5, 7, 2, 5, 9, 3, 4, 0, 6, 6, 9, 1, 1, 3, 3, 7, 8, 5, 0, 0, 6, 1, 9, 2, 3, 1, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Below regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the series:

For any number b, let S=sum((-1)^k (k^(1/k)-b), k=1..infinity) = 1/2*(b+2M-1) then b=2S+M2. Obviously, when b = M2 then S = 0, so sum((-1)^k*(k^(1/k)-M2), k=1..infinity) = 0 and limit(sum((-1)^m*(m^(1/m)-M2), m =1..2*N), N= infinity) = M = limit(sum((-1)^m*(M2-m^(1/m)), m=1..2*N-1), N=infinity).

LINKS

Table of n, a(n) for n=0..104.

Leonhard Paul Euler, Likewise, Euler shows that the sum of the divergent series 1-2+3-4+5 + etc. is 1/4.

Marvin Ray Burns, Author's original inquiry

Wolfram Alpha, 1-2*MRB constant

EXAMPLE

1-2*MRB constant = 0.624280715...

MATHEMATICA

digits = 105; 1-2*NSum[ (-1)^n*((n^(1/n)) - 1), {n, 1, Infinity}, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"] // RealDigits[#, 10, digits]& // First (* Jean-Fran├žois Alcover, Feb 15 2013 *)

CROSSREFS

Cf. A037077.

Sequence in context: A198502 A244858 A064925 * A084945 A010493 A175286

Adjacent sequences:  A173270 A173271 A173272 * A173274 A173275 A173276

KEYWORD

nonn,cons

AUTHOR

Marvin Ray Burns, Feb 14 2010, Feb 24 2010, Mar 05 2010

EXTENSIONS

Edited by Marvin Ray Burns, Apr 15 2010, Mar 04 2013

STATUS

approved

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Last modified March 18 12:10 EDT 2019. Contains 321283 sequences. (Running on oeis4.)