In[]:=
​
The MRB constant is related to the following alternating series:
S
n
=

n∞
n
∑
k=1
k
(-1)
1/k
k
==-1+2
2
-
1/3
3
-
1/5
5
+
1/6
6
-
1/n
n
==DNE
​The divergent sequence of its partial sums has two accumulation points with an upper limiting value or
, CMRB+0, and a
, CMRB - 1 This is observed and determined in the following Wolfram code and exploration .
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{l,li,s,i,W,R,DL,Dm,mm,Dp,PL,P,S,Rf,tx}="
=","
=","
n
∑
k=1
1
k
k
",Infinity,WolframAlpha,Rationalize,DiscreteMaxLimit,DiscreteMinLimit,"
+",DiscretePlot,PlotLegends,PlotLabel,StringJoin,Riffle,"f,​
and
with s. ( <> is a place holder)";
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m=W["MRB constant",{{"DecimalApproximation",1},"ComputableData"}];
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inf:=210^10;
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f[x_]:=x^(1/x);
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fi={{l,mm,R[DL[NSum[Re[(-1)^kf[k]],{k,1,inf}],ni]-m,10^-8]}};​​
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se={{li,mm,R[Dm[NSum[Re[(-1)^kf[k]],{k,1,inf+1}],ni]-m,10^-8]}};
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one=S[S[Rf[fi," "]]]//Quiet;
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two=S[Rf[se," "]]//Quiet;
In[]:=
Dp[{m,Sum[(-1`)^kf[k],{k,1,x}],m-1},{x,1,20},P->tx,PL->Placed[{one,s,two},Right],Axes{False,True}]
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