>>626
https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant
Euler-Mascheroni constant
Series expansions
In general,
γ＝lim(n→∞)1+1/2+1/3+…+1/n-log(n+α)≡lim(n→∞)γn(α)
for any α > −n .
However, the rate of convergence of this expansion depends significantly on α .
In particular, γn(1/2) exhibits much more rapid convergence than the conventional expansion γn(0).[7][8]
This is because
1/{2(n+1)} < γn(0) - γ < 1/(2n)
while
1/{24(n+1)^2} < γn(1/2) < 1/{24(n)^2}
Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.
(引用終わり)

γn(1/2)をやってみた（＾＾
オイラーγ およそ0.57721566490

n Σ1/n ln(n+1/2) Σ1/n-ln(n+1/2) [Σ1/n] [ln(n+1/2)] [Σ1/n]-[ln(n++1/2)] [1-[Σ1/n]-[ln(n++1/2)]]
1 1 0.405465108 0.594534892 0 -0.594534892 0.594534892 0.594534892
2 1.5 0.916290732 0.583709268 0.5 0.916290732 -0.416290732 0.583709268
3 1.833333333 1.252762968 0.580570365 0.833333333 0.252762968 0.580570365 0.580570365

10 2.928968254 2.351375257 0.577592997 0.928968254 0.351375257 0.577592997 0.577592997

20 3.597739657 3.020424886 0.577314771 0.597739657 0.020424886 0.577314771 0.577314771

25 3.815958178 3.238678452 0.577279726 0.815958178 0.238678452 0.577279726 0.577279726

1000 7.485470861 6.908255154 0.577215707 0.485470861 0.908255154 -0.422784293 0.577215707

5000 9.094508853 8.517293186 0.577215667 0.094508853 0.517293186 -0.422784333 0.577215667

8000 9.564474984 8.987259319 0.577215666 0.564474984 0.987259319 -0.422784334 0.577215666

9000 9.682251076 9.10503541 0.577215665 0.682251076 0.10503541 0.577215665 0.577215665

10000 9.787606036 9.210390371 0.577215665 0.787606036 0.210390371 0.577215665 0.577215665