​​TheMRBconstant​​=m=
∞
∑
n=1
n
(-1)

n
n
-1..​​mshort==
1256012054
∑
n=1
n
(-1)

n
n
-1​​AGM=ArithmeticGeometricMean;​​a=29//Range//Prime//Total;​​b=176//Range//Prime//Total;​​c=85//Range//Prime//Total
​​
m-1-

π
-
7
5
AGM[1,
2
]==-8.3403688765*
-9
10
​​m-1-

π
-
7
5
AGM1,
2
-
-1
(a-58)
(b-24)
=-2.5075367768
-17
10
​​​​
mshort-Log
(7-)(25+28)
95-2(14-23)
==-1.3640093462*
-17
10
​
​
mshort-1-

π
-
7
5
AGM[1,
2
]==-7.4188344824*
-18
10
​​
mshort-1+

π
-
7
5
AGM[1,
2
]-
125
-15
10
c+9
​
​
=3.4308286256574*
-24
10
​​
In[]:=
m=NSum[
n
(-1)
(
1/n
n
-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->60];
In[]:=
mshort=NSum[
n
(-1)
(
1/n
n
-1),{n,1,1256012054},Method->"AlternatingSigns",WorkingPrecision->60];
In[]:=
AGM=ArithmeticGeometricMean;
​
In[]:=
m-1-

π
-
7
5
AGM[1,
2
]
Out[]=
-8.3403688765533653663993629578547438005064645888310×
-9
10
In[]:=
a=29//Range//Prime//Total;​​b=176//Range//Prime//Total;
In[]:=
m-1-

π
-
7
5
AGM1,
2
-
-1
(a-58)
(b-24)
Out[]=
-2.50753677681920618556766681514593775852558×
-17
10
In[]:=
mshort-Log
(7-)(25+28)
95-2(14-23)

Out[]=
-1.36400934628190528933030628716734811515857×
-17
10
In[]:=
Nmshort-1+

π
-
7
5
AGM[1,
2
],30
Out[]=
-7.41883448240064610946690080073×
-18
10
In[]:=
mshort-1+

π
-
7
5
AGM[1,
2
]-
125
-15
10
c+9
Out[]=
3.4308286256574035366139518565446552×
-24
10