MRB Constant

Author

Eric W. Weisstein
January 21, 2020

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/MRBConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MRBConstant.html.

©2020 Wolfram Research, Inc. except for portions noted otherwise

Value

N[{#,#-1},7]&@0.18785964246206712024851793318924735092708549397887651376235100430862463221699

Plot

ListPlot[l=Accumulate[Table[(((-1))^k)(k^(1/k)),{k,100}]],Joined->True,PlotStyle->Red,AxesLabel->{n,(Subscript[s, n])}]

Computation

NSum

NSum[(-1)^k (k^(1/k)-1),{k,1,∞},WorkingPrecision->100,NSumTerms->10000]//Timing

(sum1=NSum[(-1)^k (k^(1/k)-1),{k,1},WorkingPrecision->100,Method->"AlternatingSigns"])//Timing

(sum2=NSum[(-1)^k (k^(1/k)-1),{k,∞},WorkingPrecision->200,Method->"AlternatingSigns"])//Timing

sum1-sum2

(sum3=NSum[(-1)^k (k^(1/k)-1),{k,∞},WorkingPrecision->500,Method->"AlternatingSigns"])//Timing

sum3-sum2

Accumulate (10^5)

l=Accumulate[Table[(((-1))^k)(k^(1./k)),{k,10^5}]];//Timing

Take[l,-2]

Subtract@@%

l=Accumulate[Table[(((-1))^k)(k^(1`20/k)),{k,10^5}]];//Timing

Take[l,-2]

Subtract@@%

Accumulate (10^6)

l=Accumulate[Table[(((-1))^k)(k^(1`20/k)),{k,10^6}]];//Timing

Take[l,-2]

Subtract@@%

Accumulate (10^7)

l=Accumulate[Table[(((-1))^k)(k^(1`20/k)),{k,10^7}]];//Timing

Take[l,-2]

Subtract@@%

Sums

(∑_k^∞)((((2 k))^(1/(2 k)))-(((2 k-1))^(1/(2 k-1))))

Sum[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,∞}]//Timing

V7

$Version

NSum[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,∞},WorkingPrecision->100]//Timing

V8

l=N[Table[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,10000}],1000];//Timing

SequenceLimit[Take[Accumulate[l],-1000]]

NSum[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,∞},Method->Fit,WorkingPrecision->2000,PrecisionGoal->50]//Timing

NSum[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,∞},WorkingPrecision->100]//Timing

NSum[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,∞},WorkingPrecision->1000]//Timing

NSum[(((2k))^(1/(2k)))-(((2k-1))^(1/(2k-1))),{k,∞},WorkingPrecision->1100]//Timing

c1=0.1878596424620671202485179331892473509270854939788765137623510043086246322169998165736147393993785827584455805564018210377238677619740521728815206017363481596325133994949943418968236231823777805307357850235131396737813087363209118554682273676020517693996073899138348303771887703512192631388094255017468410097658730034347696873397655781443120894693194712100636147064244526848122674511288728882658064808231547612299756920376779718193511477473250314766777045991076955526997340683049856467523946491225286134920160709827434796537298815363239486823939987837663315235095404244472976894915012805831105067907892484888087250780048466084250515893725300910594835889617631312675172564132828659010435691136909483063447946839567095092060814770182153355325333280728499910375391644784566573621366432479741419010485179156083556517054498440794719172631772624186510388038317948563748972441575056800158068114776756418446708547435842192740905591477858913332475981991235558299217558780817774792550197437310877824344559346198104961959311102964694`999.7786870798361;
c2=0.18785964246206712024851793318924735092708549397887651376235100430862463221699981657361473939937858275844558055640182103772386776197405217288152060173634815963251339949499434189682362318237778053073578502351313967378130873632091185546822736760205176939960738991383483037718877035121926313880942550174684100976587300343476968733976557814431208946931947121006361470642445268481226745112887288826580648082315476122997569201489047448618705163414692424946726106717910537861340062932639308780111675130742783339631629449167046397685596207222011059505827075354729838800156110584435050853648218425600618340605528696695095548406286782704681080137511732162702590780868197782776683737173112161642991772444056247358430747634368105818443697660671496733387936544987472179206333635344920965085931514677959063473870884473020438334144966139866199512325928751375331681038072022972777073593547318628085416058793029876376956397743189259462505273054971305450144953391808666989000566610844504253784405313554332609328856301444913052612340672220700949718440585749471851084063015302608758181795667526781513390223876842990297917978554646444107549758046658355134387`1099.7796633864518;

c1-c2

(∑_m^∞)(((-1))^m) ((m^(1/m))-1)

Sum[(((-1))^m)((m^(1/m))-1),{m,∞}]//Timing

NSum[(((-1))^m)((m^(1/m))-1),{m,∞},WorkingPrecision->1000,PrecisionGoal->200]//Timing

(∑_n^∞)(∑_k^(n+1))(((-1))^k) (k^(1/k)) (2^(-(n+1))) (Binomial[n, k-1])

Sum[(((-1))^k)Binomial[n,k-1](k^(1/k)),{k,n+1}]

Sum[(2^(-(n+1)))(((-1))^k)Binomial[n,k-1](k^(1/k)),{n,10},{k,n+1}]//N

Sum[(2^(-(n+1)))(((-1))^k)Binomial[n,k-1](k^(1/k)),{n,∞},{k,n+1}]

NSum[(2^(-(n+1)))(((-1))^k)Binomial[n,k-1](k^(1/k)),{n,∞},{k,n+1}]//Timing

-(∑_(m=1)^∞)((((-1))^m)/(m!)) Derivative[m][η](m)

From Crandall, unnumbered eqn

Table[Derivative[m][DirichletEta][m],{m,3}]//Column

c1=Limit[((((-1))^m)/(m!))Derivative[m][DirichletEta][x]/.m->1,x->1]

-(c1+Sum[((((-1))^m)/(m!))Derivative[m][DirichletEta][m],{m,2,20}])//N

-(∑_m^∞)((c(m) )/(m!))Derivative[m][η](0)

From Crandall, eqn (44)

FunctionExpand[DirichletEta[s]]

c[j_]:=Sum[Binomial[j,d](((-1))^d)(d^(j-d)),{d,j}]

c[j]

-Sum[((c[m])/(m!))Derivative[m][DirichletEta][0],{m,15}]//N

-Sum[((c[m])/(m!))Derivative[m][(1-2^(1-#))Zeta[#]&][0],{m,15}]//N

-NSum[((c[m])/(m!))Derivative[m][(1-2^(1-#))Zeta[#]&][0],{m,15}]

Integrals

(∫_0^∞)Im((((1+i t))^(1/(1+i t)))) csch(π t)dt

Marvin Burns
Jan 21, 2020

In Mathematics in the Stack Exchange Network, an internet scholar going by the moniker Dark Malthorp presented a beautiful integral for the MRB constant: the integral of Im[(1 + I t)^(1/(1 + I t))]/Sinh[Pi t] for t>0.

FullSimplify[Im[(1+I t)^(1/(1+I t))]/Sinh[Pi t]]

FunctionExpand[%]

ComplexExpand[Im[(((1+I t))^(1/(1+I t)))],TargetFunctions->{Re,Im}]

FullSimplify[%]

NIntegrate[Csch[π t] Im[(((1+I t))^(1/(1+I t)))],{t,0,∞}]

NIntegrate[Csch[π t] Im[(((1+I t))^(1/(1+I t)))],{t,0,∞},WorkingPrecision->100]