Approximating n! 10/7

A first approximation is to compare n! to nⁿ for a whole series of values of n, this is done by the function ratio1, whose m file is here:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function res=ratio(nsim);
res=zeros(1,nsim);

     for i=(1:nsim)
       res(i)=(prod(1:i))/(i^i);
      end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Output from ratio for 10 first integers:
>> ratio(10)     
ans =
  Columns 1 through 7 
    1.0000    0.5000    0.2222    0.0938    0.0384    0.0154    0.0061
  Columns 8 through 10 
    0.0024    0.0009    0.0004
%These become extremely small, so we consider
%the logs to see what is happening

>> log(ratio(100))
ans =
  Columns 1 through 7 
         0   -0.6931   -1.5041   -2.3671   -3.2597   -4.1713   -5.0962
  Columns 8 through 14 
   -6.0309   -6.9732   -7.9214   -8.8745   -9.8317  -10.7922  -11.7556
  Columns 15 through 21 
  -12.7215  -13.6896  -14.6596  -15.6312  -16.6045  -17.5790  -18.5548
  Columns 22 through 28 
  -19.5318  -20.5097  -21.4886  -22.4683  -23.4488  -24.4301  -25.4120
  Columns 29 through 35 
  -26.3945  -27.3777  -28.3614  -29.3456  -30.3303  -31.3154  -32.3010
  Columns 36 through 42 
  -33.2870  -34.2734  -35.2601  -36.2471  -37.2345  -38.2222  -39.2102
  Columns 43 through 49 
  -40.1985  -41.1871  -42.1759  -43.1649  -44.1542  -45.1437  -46.1335
  Columns 50 through 56 
  -47.1234  -48.1135  -49.1038  -50.0943  -51.0850  -52.0759  -53.0669
  Columns 57 through 63 
  -54.0581  -55.0494  -56.0409  -57.0325  -58.0243  -59.0162  -60.0082
  Columns 64 through 70 
  -61.0003  -61.9926  -62.9850  -63.9775  -64.9701  -65.9628  -66.9556
  Columns 71 through 77 
  -67.9485  -68.9416  -69.9347  -70.9279  -71.9212  -72.9146  -73.9081
  Columns 78 through 84 
  -74.9016  -75.8953  -76.8890  -77.8828  -78.8767  -79.8706  -80.8647
  Columns 85 through 91 
  -81.8588  -82.8529  -83.8471  -84.8414  -85.8358  -86.8302  -87.8247
  Columns 92 through 98 
  -88.8193  -89.8139  -90.8085  -91.8032  -92.7980  -93.7928  -94.7877
  Columns 99 through 100 
  -95.7827  -96.7776

%There seems to be a linear dependence on $n$ here,
%we try dividing by n at each step:

>> log(ratio(100))./(1:100)
ans =
  Columns 1 through 7 
         0   -0.3466   -0.5014   -0.5918   -0.6519   -0.6952   -0.7280
  Columns 8 through 14 
   -0.7539   -0.7748   -0.7921   -0.8068   -0.8193   -0.8302   -0.8397
  Columns 15 through 21 
   -0.8481   -0.8556   -0.8623   -0.8684   -0.8739   -0.8790   -0.8836
  Columns 22 through 28 
   -0.8878   -0.8917   -0.8954   -0.8987   -0.9019   -0.9048   -0.9076
  Columns 29 through 35 
   -0.9102   -0.9126   -0.9149   -0.9170   -0.9191   -0.9210   -0.9229
  Columns 36 through 42 
   -0.9246   -0.9263   -0.9279   -0.9294   -0.9309   -0.9322   -0.9336
  Columns 43 through 49 
   -0.9348   -0.9361   -0.9372   -0.9384   -0.9395   -0.9405   -0.9415
  Columns 50 through 56 
   -0.9425   -0.9434   -0.9443   -0.9452   -0.9460   -0.9468   -0.9476
  Columns 57 through 63 
   -0.9484   -0.9491   -0.9498   -0.9505   -0.9512   -0.9519   -0.9525
  Columns 64 through 70 
   -0.9531   -0.9537   -0.9543   -0.9549   -0.9554   -0.9560   -0.9565
  Columns 71 through 77 
   -0.9570   -0.9575   -0.9580   -0.9585   -0.9589   -0.9594   -0.9598
  Columns 78 through 84 
   -0.9603   -0.9607   -0.9611   -0.9615   -0.9619   -0.9623   -0.9627
  Columns 85 through 91 
   -0.9630   -0.9634   -0.9638   -0.9641   -0.9644   -0.9648   -0.9651
  Columns 92 through 98 
   -0.9654   -0.9657   -0.9660   -0.9663   -0.9666   -0.9669   -0.9672
  Columns 99 through 100 
   -0.9675   -0.9678
%Looks like this tends to -1, (roughly)

This means:

$$log(\frac{n!}{n^n}) \longrightarrow -n$$

Or

$$\frac{n!}{n^n} \longrightarrow exp(-n) \Rightarrow n!\longrightarrow (\frac{n}{e})^n$$

In fact there is a term in n left because the ratio of this approximation to n! still increases (in fact like $\sqrt{n}$.

The true approximation is given by Stirling's formula which uses the notion of asymptotically equivalent.

Definition:
Two sequences a_k and b_k are said to be asymptotically equivalent if

$$lim_{k\longrightarrow \infty} \frac{a_k}{b_k} = 1$$

Theorem 1:(Stirling's Formula)
Factorial n is asymptotically equivalent to the sequence defined as: $n^ne^{-n}\sqrt{2\pi n}$.

$$n!\sim (\frac{n}{e})^n \sqrt{2\pi n}$$

Here is the file stirling.m:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function res=stirling(nsim)
res=zeros(1,nsim);

     for i=(1:nsim)
         res(i)=(i/exp(1))^i*i^(1/2)*sqrt(2*pi)/prod(1:i);
      end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This gives the following series of ratios showing that the approximation gets progressively better:

>> stirling(100)
ans =
  Columns 1 through 7 
    0.9221    0.9595    0.9727    0.9794    0.9835    0.9862    0.9882
  Columns 8 through 14 
    0.9896    0.9908    0.9917    0.9925    0.9931    0.9936    0.9941
  Columns 15 through 21 
    0.9945    0.9948    0.9951    0.9954    0.9956    0.9958    0.9960
  Columns 22 through 28 
    0.9962    0.9964    0.9965    0.9967    0.9968    0.9969    0.9970
  Columns 29 through 35 
    0.9971    0.9972    0.9973    0.9974    0.9975    0.9976    0.9976
  Columns 36 through 42 
    0.9977    0.9978    0.9978    0.9979    0.9979    0.9980    0.9980
  Columns 43 through 49 
    0.9981    0.9981    0.9981    0.9982    0.9982    0.9983    0.9983
  Columns 50 through 56 
    0.9983    0.9984    0.9984    0.9984    0.9985    0.9985    0.9985
  Columns 57 through 63 
    0.9985    0.9986    0.9986    0.9986    0.9986    0.9987    0.9987
  Columns 64 through 70 
    0.9987    0.9987    0.9987    0.9988    0.9988    0.9988    0.9988
  Columns 71 through 77 
    0.9988    0.9988    0.9989    0.9989    0.9989    0.9989    0.9989
  Columns 78 through 84 
    0.9989    0.9989    0.9990    0.9990    0.9990    0.9990    0.9990
  Columns 85 through 91 
    0.9990    0.9990    0.9990    0.9991    0.9991    0.9991    0.9991
  Columns 92 through 98 
    0.9991    0.9991    0.9991    0.9991    0.9991    0.9991    0.9992
  Columns 99 through 100 
    0.9992    0.9992