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First 100 partial sums of $(-1)^{k}(k^{1/k}-1)$

The MRB constant is a mathematical constant, with decimal expansion 0.187859… (sequence A037077 in the OEIS). The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999.^[1] Burns had initially called the constant "rc" for root constant^[2] but, at Simon Plouffe's suggestion, the constant was renamed the 'Marvin Ray Burns's Constant', or "MRB constant".^[3]

The MRB constant is defined as the upper limit of the partial sums^[4]^[5]^[6]^[7]^[8]^[9]^[10]

s_{n}=\sum _{k=1}^{n}(-1)^{k}k^{1/k}

As $n$ grows to infinity, the sums have upper and lower limit points of −0.812140… and 0.187859…, separated by an interval of length 1. The constant can also be explicitly defined by the following infinite sums:^[4]

0.187859\ldots =\sum _{k=1}^{\infty }(-1)^{k}(k^{1/k}-1)=\sum _{k=1}^{\infty }\left((2k)^{1/(2k)}-(2k-1)^{1/(2k-1)}\right).

The constant relates to the divergent series:

\sum _{k=1}^{\infty }(-1)^{k}k^{1/k}.

There is no known closed-form expression of the MRB constant,^[11] nor is it known whether the MRB constant is algebraic, transcendental or even irrational.

References

^ Plouffe, Simon. "mrburns". Retrieved 12 January 2015.
^ Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.
^ Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.
^ ^a ^b Weisstein, Eric W. "MRB Constant". MathWorld.
^ Mathar, Richard J. (2009). "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844 [math.CA].
^ Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original on April 30, 2013. Retrieved 16 January 2015.
^ (sequence A037077 in the OEIS)
^ (sequence A160755 in the OEIS)
^ (sequence A173273 in the OEIS)
^ Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.

External links

Official site of M.R. Burns, constant's namesake and discoverer

[1] Plouffe, Simon. "mrburns". Retrieved 12 January 2015.

[2] Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.

[3] Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.

[Weisstein-4] Weisstein, Eric W. "MRB Constant". MathWorld.

[5] Mathar, Richard J. (2009). "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844 [math.CA].

[6] Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original on April 30, 2013. Retrieved 16 January 2015.

[7] (sequence A037077 in the OEIS)

[8] (sequence A160755 in the OEIS)

[9] (sequence A173273 in the OEIS)

[10] Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.

[11] Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.

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@@ Line 1: / Line 1: @@
+{{Short description|Mathematical constant described by Marvin Ray Burns}}
-{{COI|date=November 2015}}
+[[File:MRB-Gif.gif|thumb|right|195px|First 100 partial sums of <math>(-1)^k (k^{1/k} - 1)</math>]]The '''MRB constant''' is a [[mathematical constant]], with decimal expansion {{nowrap|0.187859…}} {{OEIS|A037077}}. The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999.<ref>{{cite web|url=http://www.plouffe.fr/simon/constants/mrburns.txt|title=mrburns|last=Plouffe|first=Simon|access-date=12 January 2015}}</ref> Burns had initially called the constant "rc" for root constant<ref>{{cite web|url=http://math2.org/mmb/thread/901|title=RC|last=Burns|first=Marvin R.|date=23 January 1999|website=math2.org|access-date=5 May 2009}}</ref> but, at [[Simon Plouffe|Simon Plouffe's]] suggestion, the constant was renamed the 'Marvin Ray Burns's Constant', or "MRB constant".<ref>{{cite web|url=http://www.plouffe.fr/simon/articles/Tableofconstants.pdf|title=Tables of Constants|last=Plouffe|first=Simon|date=20 November 1999|publisher=Laboratoire de combinatoire et d'informatique mathématique|access-date=5 May 2009}}</ref>
-[[File:MRB messy.gif|thumb|right|195px|Marvin R. Burns, the constant's author, in 1999]]
+The MRB constant is defined as the [[upper limit]] of the [[partial sums]]<ref name="Weisstein" /><ref>{{cite arXiv|eprint=0912.3844|first=Richard J.|last=Mathar|title=Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity|year=2009|class=math.CA}}</ref><ref>{{cite web|url=http://www.perfscipress.com/papers/UniversalTOC25.pdf|title=Unified algorithms for polylogarithm, L-series, and zeta variants|last=Crandall|first=Richard|publisher=PSI Press|archive-url=https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf|archive-date=April 30, 2013|url-status=usurped|access-date=16 January 2015}}</ref><ref>{{OEIS|id=A037077}}</ref><ref>{{OEIS|id=A160755}}</ref><ref>{{OEIS|id=A173273}}</ref><ref>{{cite web|url=http://www.bitman.name/math/article/962|title=MRB (costante)|last=Fiorentini|first=Mauro|website=bitman.name|language=italian|access-date=14 January 2015}}</ref>
-The '''MRB constant,''' named after Marvin Ray Burns, is a [[mathematical constant]] for which no [[closed-form expression]] is known. It is not known whether the MRB constant is [[algebraic number|algebraic]], [[transcendental number|transcendental]], or even [[irrational number|irrational]].
-The numerical value of MRB constant, truncated to 6 [[decimal|decimal places]], is
-:{{nowrap|0.187859…}} {{OEIS|A037077}}.
-==Definition==
-[[File:MRB-Gif.gif|thumb|right|195px|First 100 partial sums of <math>(-1)^k (k^{1/k} - 1)</math>]]
-The MRB constant is related to the following [[divergent series]]:
-: <math>\sum_{k=1}^{\infty} (-1)^k k^{1/k}.</math>
-Its [[partial sum]]s
 : <math>s_n = \sum_{k=1}^n (-1)^k k^{1/k}</math>
-are bounded so that their [[limit point]]s form an [[interval (mathematics)|interval]] [−0.812140…,0.187859…] of length 1. The [[upper limit]] point 0.187859… is what is known as the MRB constant.<ref name="Weisstein"/><ref>{{cite arXiv |last=Mathar |first=Richard J. |title=Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity |arxiv=0912.3844}}</ref><ref>{{cite web |last=Crandall |first=Richard |title=Unified algorithms for polylogarithm, L-series, and zeta variants |url=http://www.perfscipress.com/papers/UniversalTOC25.pdf |publisher=PSI Press |accessdate=16 January 2015 |deadurl=yes |archiveurl=https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf |archivedate=April 30, 2013}}</ref><ref>{{OEIS|id=A037077}}</ref><ref>{{OEIS|id=A160755}}</ref><ref>{{OEIS|id=A173273}}</ref><ref>{{cite web |last=Fiorentini |first=Mauro |title=MRB (costante) |url=http://www.bitman.name/math/article/962 |website=bitman.name |accessdate=14 January 2015 |language=italian}}</ref>
-The MRB constant can be explicitly defined by the following infinite sums:<ref name="Weisstein">{{MathWorld |title=MRB Constant |urlname=MRBConstant}}</ref>
+As <math>n</math> grows to infinity, the sums have [[Limit inferior and limit superior|upper and lower limit points]] of −0.812140… and 0.187859…, separated by an [[interval (mathematics)|interval]] of length 1. The constant can also be explicitly defined by the following infinite sums:<ref name="Weisstein">{{MathWorld |title=MRB Constant |urlname=MRBConstant}}</ref>
 : <math>0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right).</math>
+The constant relates to the [[divergent series]]:
-There is no known [[closed-form expression]] of the MRB constant.<ref>{{cite book |title=Mathematical Constants |last=Finch |first=Steven R. |year=2003 |publisher=[[Cambridge University Press]] |location=[[Cambridge, England]] |isbn=0-521-81805-2 |page=450}}</ref>
+:<math>\sum_{k=1}^{\infty} (-1)^k k^{1/k}.</math>
+There is no known [[closed-form expression]] of the MRB constant,<ref>{{cite book|title=Mathematical Constants|url=https://archive.org/details/mathematicalcons0000finc|url-access=registration|last=Finch|first=Steven R.|publisher=[[Cambridge University Press]]|year=2003|isbn=0-521-81805-2|location=[[Cambridge, England]]|page=[https://archive.org/details/mathematicalcons0000finc/page/450 450]}}</ref> nor is it known whether the MRB constant is [[algebraic number|algebraic]], [[Transcendental number|transcendental]] or even [[irrational number|irrational]].
-==History==
-Marvin Ray Burns published his discovery of the constant in 1999.<ref>{{cite web |last=Burns |first=Marvin |title=mrburns |url=http://www.plouffe.fr/simon/constants/mrburns.txt |publisher=Simeon Plouffe |accessdate=12 January 2015}}</ref> The discovery is a result of a "math [[wikt:binge|binge]]" that started in the spring of 1994.<ref>{{cite web |last=Burns |first=Marvin R. |title=Captivity's Captor: Now is the Time for the Chorus of Conversion |url=https://oncourse.iu.edu/access/content/user/marburns/Filemanager_Public_Files/final1.doc |date=12 April 2002 |publisher=[[Indiana University]] |accessdate=5 May 2009}}</ref> Before verifying with colleague [[Simon Plouffe]] that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant.<ref>{{cite web |last=Burns |first=Marvin R. |title=RC |url=http://math2.org/mmb/thread/901 |date=23 January 1999 |website=math2.org |accessdate=5 May 2009}}</ref> At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.<ref>{{cite web |last=Plouffe |first=Simon |title=Tables of Constants |url=http://www.plouffe.fr/simon/articles/Tableofconstants.pdf |date=20 November 1999 | publisher=Laboratoire de combinatoire et d'informatique mathématique |accessdate=5 May 2009}}</ref>
 ==References==
@@ Line 28: / Line 20: @@
 ==External links==
 {{Portal|Mathematics}}
-* [http://marvinrayburns.com/ Official site of M.R. Burns, constant's author]
+* [http://marvinrayburns.com/ Official site of M.R. Burns, constant's namesake and discoverer]
 [[Category:Mathematical constants]]