■スターリングの公式の図形的証明?(その32)

 (その30)では

  n^n/n!≧(8/π)^n/2         (n≧30)

  n^n/n!≧(8/π)^n・√π/(2n)   (n≧18)

としたが,実際に調べたのは

  e^n/√(2πn)≧(8/π)^n/2         (n≧30)

  e^n/√(2πn)≧(8/π)^n・√π/(2n)   (n≧18)

であって,肝心の

  L0=ln(n^n/n!)

が計算できなかった.

 畏友・阪本ひろむ氏に計算してもらったL0の値と

  L1=ln(e^n/√(2πn))

  L2=ln(2^n/2)

  L3=ln((8/π)^n/2)

  L4=ln((8/π)^n・√π/(2n))

の計算結果も併せて以下に掲げるが,

  n^n/n!≧(8/π)^n/2         (n≧30)

  n^n/n!≧(8/π)^n・√π/(2n)   (n≧18)

という結論は変わらない.

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n L0 L1 L2 L3 L4

1 0. 0.0810615 0. 0.241564 1.1605

2 0.693147 0.734488 0.693147 1.17628 1.74864

3 1.50408 1.53176 1.38629 2.11099 2.48062

4 2.36712 2.38791 2.07944 3.0457 3.27149

5 3.2597 3.27634 2.77259 3.98041 4.09463

6 4.17131 4.18518 3.46574 4.91512 4.93818

7 5.09621 5.10811 4.15888 5.84983 5.79582

8 6.03093 6.04134 4.85203 6.78455 6.66376

9 6.97319 6.98245 5.54518 7.71926 7.53958

10 7.92144 7.92977 6.23832 8.65397 8.42162

11 8.87454 8.88211 6.93147 9.58868 9.30867

12 9.83167 9.83861 7.62462 10.5234 10.1999

13 10.7922 10.7986 8.31777 11.4581 11.0946

14 11.7556 11.7615 9.01091 12.3928 11.9922

15 12.7215 12.727 9.70406 13.3275 12.8924

16 13.6896 13.6948 10.3972 14.2622 13.7949

17 14.6596 14.6645 11.0904 15.197 14.6993

18 15.6312 15.6359 11.7835 16.1317 15.6054

19 16.6045 16.6088 12.4766 17.0664 16.5131

20 17.579 17.5832 13.1698 18.0011 17.4222

21 18.5548 18.5588 13.8629 18.9358 18.3325

22 19.5318 19.5355 14.5561 19.8705 19.2439

23 20.5097 20.5133 15.2492 20.8052 20.1564

24 21.4886 21.492 15.9424 21.7399 21.0698

25 22.4683 22.4716 16.6355 22.6746 21.9841

26 23.4488 23.452 17.3287 23.6094 22.8992

27 24.4301 24.4331 18.0218 24.5441 23.8151

28 25.412 25.415 18.715 25.4788 24.7316

29 26.3945 26.3974 19.4081 26.4135 25.6488

30 27.3777 27.3805 20.1013 27.3482 26.5665

31 28.3614 28.3641 20.7944 28.2829 27.4849

32 29.3456 29.3482 21.4876 29.2176 28.4037

33 30.3303 30.3328 22.1807 30.1523 29.323

34 31.3154 31.3179 22.8739 31.087 30.2428

35 32.301 32.3034 23.567 32.0218 31.163

36 33.287 33.2893 24.2602 32.9565 32.0837

37 34.2734 34.2756 24.9533 33.8912 33.0047

38 35.2601 35.2623 25.6464 34.8259 33.926

39 36.2471 36.2493 26.3396 35.7606 34.8478

40 37.2345 37.2366 27.0327 36.6953 35.7698

41 38.2222 38.2243 27.7259 37.63 36.6922

42 39.2102 39.2122 28.419 38.5647 37.6148

43 40.1985 40.2005 29.1122 39.4995 38.5378

44 41.1871 41.189 29.8053 40.4342 39.461

45 42.1759 42.1777 30.4985 41.3689 40.3845

46 43.1649 43.1667 31.1916 42.3036 41.3082

47 44.1542 44.156 31.8848 43.2383 42.2322

48 45.1437 45.1455 32.5779 44.173 43.1564

49 46.1335 46.1352 33.2711 45.1077 44.0808

50 47.1234 47.125 33.9642 46.0424 45.0054

51 48.1135 48.1151 34.6574 46.9771 45.9302

52 49.1038 49.1054 35.3505 47.9119 46.8552

53 50.0943 50.0959 36.0437 48.8466 47.7804

54 51.085 51.0866 36.7368 49.7813 48.7057

55 52.0759 52.0774 37.4299 50.716 49.6313

56 53.0669 53.0684 38.1231 51.6507 50.557

57 54.0581 54.0595 38.8162 52.5854 51.4828

58 55.0494 55.0508 39.5094 53.5201 52.4088

59 56.0409 56.0423 40.2025 54.4548 53.335

60 57.0325 57.0339 40.8957 55.3896 54.2613

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