19/03/22 14:20:17.81 WSdp8+VY.net
>>626
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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