Euler–Mascheroni constant analog to the MRB constant
Euler–Mascheroni constant analog to the MRB constant
Compare the Euler–Mascheroni constant, γ, with unknown rationality but by MathWorld possessing the following bounds,
with the an analog, , of the MRB constant and its integrated analog,
γ
CMRB
∞
∑
k=1
k
(-1)
1/k
k
γ
CMRB
∞
∫
1
k
(-1)
1/k
k
which is well-approximated by combinations of rational numbers and e:
NSum[(-1)^k(k^(1/k)-1),{k,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->30]-Abs[NIntegrate[(-1)^k(k^(1/k)-1),{k,1,InfinityI},WorkingPrecision->30]+2/(IPi)]
Out[]=
-0.49979272646562724956073343752
%+1/2-1/5000
Out[]=
7.27353437275043926656248×
-6
10
NSum[(-1)^k(k^(1/k)-1),{k,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->30]-Abs[NIntegrate[(-1)^k(k^(1/k)-1),{k,1,InfinityI},WorkingPrecision->20]]
Out[]=
0.102688223461358397059
%-
9/5+E
44
Out[]=
8.7289187165×
-11
10