MaplePrimes - comments on Post, MRB Constant G

MaplePrimes - comments on Post, MRB Constant G http://www.mapleprimes.com/posts/35691-MRB-Constant-G en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 21:28:08 GMT Wed, 10 Jun 2026 21:28:08 GMT The latest comments added to the Post, MRB Constant G http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, MRB Constant G http://www.mapleprimes.com/posts/35691-MRB-Constant-G 1/3(2*MRB constant-1) http://www.mapleprimes.com/posts/35691-MRB-Constant-G?ref=Feed:MaplePrimes:MRB Constant G:Comments#comment44727 In the blog <a href="http://www.mapleprimes.com/blog/marvinrayburns/mrbconstantf">MRB Constant F</a> we looked at the sequences of series of the form sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity) and found an a, that Maple says is a zero for sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity), namely 1-2*MRB constant. Now lets look at what happens if we let a=M where M=sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity), the MRB constant, and apply the generating function, M[p+1] = sum((-1)^m*(m^(1/m)+M[p]), m = 1 .. infinity) for several iterations. To begin to see the hyper-geometric application of each step of this iteration follow the link at <a href="http://www.mapleprimes.com/blog/marvinrayburns/mrbconstante">MRB Constant E</a>. In Maple, Digits := 30; M := evalf(sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity)); for a to 60 do M := evalf(sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity)); print(evalf(M)) end do; (giving an end result of -.208093571691955...). According to Wolfram Alpha, -0.208093571691955253167654710630484513296064603399907475885... is 1/3(2*MRB constant-1). So it appears that the infinite iteration of M[0]=sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity) and M[p+1] = sum((-1)^n*(n^(1/n)+M[p]), n = 1 .. infinity) gives M[p=infinity] = 1/3(2*MRB constant-1). Note from ?evalf,Sum help-page: In the case of infinite sums, Levin's u-transform is used, which has the additional effect that sums that formally diverge may return a result that can be interpreted as evaluation of the analytic extension of the series for the sum. The latest comments added to the Post, MRB Constant G 44727 Tue, 02 Mar 2010 05:09:47 Z Marvin Ray Burns Marvin Ray Burns Infinite iteration of M[p] = -1/3MRB2 http://www.mapleprimes.com/posts/35691-MRB-Constant-G?ref=Feed:MaplePrimes:MRB Constant G:Comments#comment44728 Remember, in the case of infinite sums, Levin's u-transform is used by Maple, which has the additional effect that sums that formally diverge may return a result that can be interpreted as evaluation of the analytic extension of the series for the sum. Since 1-2M = -1*(2M-1), we can let M=MRB constant=sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity) and  MRB2=1-2M=0.62428071507586575950296413189145353988819381019972242765599..., so the zero for sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity) is MRB2, and let M[0]=sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity) and M[p+1] = sum((-1)^n*(n^(1/n)+M[p]), n = 1 .. infinity), then Limit(M[p],p=infinity) = -1/3MRB2. The latest comments added to the Post, MRB Constant G 44728 Tue, 02 Mar 2010 07:32:20 Z Marvin Ray Burns Marvin Ray Burns Short list of recently discovered identities http://www.mapleprimes.com/posts/35691-MRB-Constant-G?ref=Feed:MaplePrimes:MRB Constant G:Comments#comment44730 Let M = MRB constant = sum((-1)^n*(n^(1/n)-1),n =1 .. infinity), and let MRB2 = 1-2M = 0.6242807150758657595... : The zero for sum((-1)^n*(n^(1/n)-a),n = 1 .. infinity) is a = MRB2. Let M4 = sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity) = 1/2*(M-1), and let M5 = sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity) = 1/2*(3*M-1). Let N = M4 + M5 = -1+2*M = -(1-2M) = -MRB2. Let M[0] = M and M[p+1] = sum((-1)^n*(n^(1/n)+M[p]),n =1 .. infinity), then Limit(M[p],p=infinity) = -1/3MRB2 = 1/3*N   In the case of infinite sums, Levin's u-transform is used, which has the additional effect that sums that formally diverge may return a result that can be interpreted as evaluation of the analytic extension of the series for the sum. The latest comments added to the Post, MRB Constant G 44730 Wed, 03 Mar 2010 04:51:41 Z Marvin Ray Burns Marvin Ray Burns sum((-1)^m*(m^(1/m)±M), m = 1 .. infinity) http://www.mapleprimes.com/posts/35691-MRB-Constant-G?ref=Feed:MaplePrimes:MRB Constant G:Comments#comment44729   Previously we found a exact form for the generating function, M[p+1] = sum((-1)^n*(n^(1/n)+M[p]), n = 1 .. infinity), which was -1/3MRB2 where MRB2 is 1-2M. As before, M is the MRB constant and Maple will be using the Levin's u-transform which can assign values to divergent series. ( Euler also found a value to the, divergent, first general infinite series:, as seen in <a href="http://www.math.dartmouth.edu/~euler/docs/translations/E352.pdf">http://www.math.dartmouth.edu/~euler/docs/translations/E352.pdf </a>.) Now let's just concentrate on the nested part of the generating function, namley, sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity), and the like summation, sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity). Through numeric evaluation it becomes rather clear that sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity) = <img alt="" src="file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot-26.png" />1/2*(M-1) and sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity) = <img alt="" src="file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot-26.png" />1/2*(3*M-1), which can be shown in Maple as follows. Digits:=70: <img alt="" src="file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot-27.png" />M := sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity): <img alt="" src="file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot-28.png" />M4:=sum((-1)^m*(m^(1/m)+evalf(M)), m = 1 .. infinity):evalf(M4-1/2*(M-1)); M5:=sum((-1)^m*(m^(1/m)-evalf(M)), m = 1 .. infinity):evalf(M5-1/2*(3*M-1)); Thier combined sum is -1+2*M = -(1-2M) = -MRB2, as seen by evalf(M4+M5-2*M+1). The latest comments added to the Post, MRB Constant G 44729 Sun, 07 Mar 2010 06:21:22 Z Marvin Ray Burns Marvin Ray Burns Give the same numeric sum http://www.mapleprimes.com/posts/35691-MRB-Constant-G?ref=Feed:MaplePrimes:MRB Constant G:Comments#comment44731 In the previous messages Maple  found a value for the divergent series M4 and M5.   Wolfram Alpha, now, gives the same numeric sums, albeit issuing a warning. Click: <a href="http://www2.wolframalpha.com/input/?i=sum%28%28-1%29^n*%28n^%281%2Fn%29%2Bevalf%28MRB+constant%2C10%29%29%2C+n+%3D+1+..+infinity%29">Let M4 = sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity) = 1/2*(M-1)</a> or <a href="http://www2.wolframalpha.com/input/?i=sum%28%28-1%29^n*%28n^%281%2Fn%29-evalf%28MRB+constant%2C10%29%29%2C+n+%3D+1+..+infinity%29">let M5 = sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity) = 1/2*(3*M-1).</a> Their combined sum was N = -MRB2, <a href="http://oeis.org/A173273"> Click here for link to published description of MRB2.</a> (If you have any additional insights please contribute to that entry.) Let M[0] = M and M[p+1] = sum((-1)^n*(n^(1/n)+M[p]),n =1 .. infinity), then N=3*Limit(M[p],p=infinity). The latest comments added to the Post, MRB Constant G 44731 Mon, 22 Mar 2010 06:31:16 Z Marvin Ray Burns Marvin Ray Burns A new theorem http://www.mapleprimes.com/posts/35691-MRB-Constant-G?ref=Feed:MaplePrimes:MRB Constant G:Comments#comment44733 Notice that M = 1/2*(M4+M5+1); which means  sum((-1)^n*(n^(1/n)-1),n =1 .. infinity) = 1/2*(sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity)+sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity)). Thus M = 1/2*(sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity)+sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity)), where M is the MRB constant. That equation  is not true if M=1. This question remains open: Is the MRB constant the only value for M such that M = 1/2*(sum((-1)^m*(m^(1/m)+M), m = 1 .. infinity)+sum((-1)^m*(m^(1/m)-M), m = 1 .. infinity)) and such that sum((-1)^n*(n^(1/n)-(1-2M)), n = 1 .. infinity) = 0? If you are new to this blog and are concerned about the divergent series: "In the case of infinite sums, Levin's u-transform is used, which has the additional effect that sums that formally diverge may return a result that can be interpreted as evaluation of the analytic extension of the series for the sum."(from Maple's help files)     <a href="http://marvinrayburns.com">marvinrayburns.com </a> The latest comments added to the Post, MRB Constant G 44733 Mon, 22 Mar 2010 07:32:15 Z Marvin Ray Burns Marvin Ray Burns