Chapter 17, Problem 17.57E

Interpretation Introduction

Interpretation:

The values of logarithm of $N!$, $N=1$ to $100$, explicitly and by using Stirling’s approximation is to be calculated and the values are to be compared. The approximate value at which the Stirling’s approximation agrees with the true value to within $1\%$ is to be stated.

Concept introduction:

Combination formula is a possible way of grouping of distinguishable objects into various subsystems. The equation for combination formula is as follows:

$C=\frac{N!}{\underset{i=1}{\overset{m}{\prod n_i!}}}$

Where,

• $N$ is the object.

If the value of $N$ is large, then the Stirling’s approximation method is used to estimate factorials. The Stirling’s approximation for large value of $N$ is,

$\ln N!\approx N\ln N-N$

Answer to Problem 17.57E

For $N\ge 90$ Stirling’s approximation agrees with the true value to within $1\%$.

Explanation of Solution

The Stirling’s approximation for large value of $N$ is,

$\ln N!\approx N\ln N-N$ …(1)

$N$$\ln N!$$N\ln N-N$

$\begin{array}{rrr}\hfill 1& \hfill 0& \hfill -1\\ \hfill 2& \hfill 0.693147& \hfill -0.61371\\ \hfill 3& \hfill 1.791759& \hfill 0.295837\\ \hfill 4& \hfill 3.178054& \hfill 1.545177\end{array}$

$\begin{array}{rrr}\hfill 5& \hfill 4.787492& \hfill 3.04719\\ \hfill 6& \hfill 6.579251& \hfill 4.750557\\ \hfill 7& \hfill 8.525161& \hfill 6.621371\\ \hfill 8& \hfill 10.6046& \hfill 8.635532\end{array}$

$\begin{array}{rrr}\hfill 9& \hfill 12.80183& \hfill 10.77502\\ \hfill 10& \hfill 15.10441& \hfill 13.02585\\ \hfill 11& \hfill 17.50231& \hfill 15.37685\\ \hfill 12& \hfill 19.98721& \hfill 17.81888\end{array}$

$\begin{array}{rrr}\hfill 13& \hfill 22.55216& \hfill 20.34434\\ \hfill 14& \hfill 25.19122& \hfill 22.9468\\ \hfill 15& \hfill 27.89927& \hfill 25.62075\\ \hfill 16& \hfill 30.67186& \hfill 28.36142\end{array}$

$\begin{array}{rrr}\hfill 17& \hfill 33.50507& \hfill 31.16463\\ \hfill 18& \hfill 36.39545& \hfill 34.02669\\ \hfill 19& \hfill 39.33988& \hfill 36.94434\\ \hfill 20& \hfill 42.33562& \hfill 39.91465\end{array}$

$\begin{array}{rrr}\hfill 21& \hfill 45.38014& \hfill 42.93497\\ \hfill 22& \hfill 48.47118& \hfill 46.00293\\ \hfill 23& \hfill 51.60668& \hfill 49.11637\\ \hfill 24& \hfill 54.78473& \hfill 52.27329\end{array}$

$\begin{array}{rrr}\hfill 25& \hfill 58.00361& \hfill 55.4719\\ \hfill 26& \hfill 61.2617& \hfill 58.71051\\ \hfill 27& \hfill 64.55754& \hfill 61.9876\\ \hfill 28& \hfill 67.88974& \hfill 65.30173\end{array}$

$\begin{array}{rrr}\hfill 29& \hfill 71.25704& \hfill 68.65158\\ \hfill 30& \hfill 74.65824& \hfill 72.03592\\ \hfill 31& \hfill 78.09222& \hfill 75.4536\\ \hfill 32& \hfill 81.55796& \hfill 78.90355\end{array}$

$\begin{array}{rrr}\hfill 33& \hfill 85.05447& \hfill 82.38475\\ \hfill 34& \hfill 88.58083& \hfill 85.89626\\ \hfill 35& \hfill 92.13618& \hfill 89.43718\\ \hfill 36& \hfill 95.71969& \hfill 93.00668\end{array}$

$\begin{array}{rrr}\hfill 37& \hfill 99.33061& \hfill 96.60396\\ \hfill 38& \hfill 102.9682& \hfill 100.2283\\ \hfill 39& \hfill 106.6318& \hfill 103.8789\\ \hfill 40& \hfill 110.3206& \hfill 107.5552\end{array}$

$\begin{array}{rrr}\hfill 41& \hfill 114.0342& \hfill 111.2565\\ \hfill 42& \hfill 117.7719& \hfill 114.9821\\ \hfill 43& \hfill 121.5331& \hfill 118.7316\\ \hfill 44& \hfill 125.3173& \hfill 122.5043\end{array}$

$\begin{array}{rrr}\hfill 45& \hfill 129.1239& \hfill 126.2998\\ \hfill 46& \hfill 132.9526& \hfill 130.1175\\ \hfill 47& \hfill 136.8027& \hfill 133.9569\\ \hfill 48& \hfill 140.6739& \hfill 137.8176\end{array}$

$\begin{array}{rrr}\hfill 49& \hfill 144.5657& \hfill 141.6992\\ \hfill 50& \hfill 148.4778& \hfill 145.6012\\ \hfill 51& \hfill 152.4096& \hfill 149.5231\\ \hfill 52& \hfill 156.3608& \hfill 153.4647\end{array}$

$\begin{array}{rrr}\hfill 53& \hfill 160.3311& \hfill 157.4255\\ \hfill 54& \hfill 164.3201& \hfill 161.4051\\ \hfill 55& \hfill 168.3274& \hfill 165.4033\\ \hfill 56& \hfill 172.3528& \hfill 169.4197\end{array}$

$\begin{array}{rrr}\hfill 57& \hfill 176.3958& \hfill 173.4539\\ \hfill 58& \hfill 180.4563& \hfill 177.5057\\ \hfill 59& \hfill 184.5338& \hfill 181.5747\\ \hfill 60& \hfill 188.6282& \hfill 185.6607\end{array}$

$\begin{array}{rrr}\hfill 61& \hfill 192.739& \hfill 189.7633\\ \hfill 62& \hfill 196.8662& \hfill 193.8823\\ \hfill 63& \hfill 201.0093& \hfill 198.0175\\ \hfill 64& \hfill 205.1682& \hfill 202.1685\end{array}$

$\begin{array}{rrr}\hfill 65& \hfill 209.3426& \hfill 206.3352\\ \hfill 66& \hfill 213.5322& \hfill 210.5172\\ \hfill 67& \hfill 217.7369& \hfill 214.7144\\ \hfill 68& \hfill 221.9564& \hfill 218.9265\end{array}$

$\begin{array}{rrr}\hfill 69& \hfill 226.1905& \hfill 223.1533\\ \hfill 70& \hfill 230.439& \hfill 227.3947\\ \hfill 71& \hfill 234.7017& \hfill 231.6503\\ \hfill 72& \hfill 238.9784& \hfill 235.92\end{array}$

$\begin{array}{rrr}\hfill 73& \hfill 243.2688& \hfill 240.2035\\ \hfill 74& \hfill 247.5729& \hfill 244.5008\\ \hfill 75& \hfill 251.8904& \hfill 248.8116\\ \hfill 76& \hfill 256.2211& \hfill 253.1357\end{array}$

$\begin{array}{rrr}\hfill 77& \hfill 260.5649& \hfill 257.473\\ \hfill 78& \hfill 264.9216& \hfill 261.8233\\ \hfill 79& \hfill 269.2911& \hfill 266.1864\\ \hfill 80& \hfill 273.6731& \hfill 270.5621\end{array}$

$\begin{array}{rrr}\hfill 81& \hfill 278.0676& \hfill 274.9504\\ \hfill 82& \hfill 282.4743& \hfill 279.351\\ \hfill 83& \hfill 286.8931& \hfill 283.7638\\ \hfill 84& \hfill 291.324& \hfill 288.1886\end{array}$

$\begin{array}{rrr}\hfill 85& \hfill 295.7666& \hfill 292.6254\\ \hfill 86& \hfill 300.2209& \hfill 297.0739\\ \hfill 87& \hfill 304.6869& \hfill 301.534\\ \hfill 88& \hfill 309.1642& \hfill 306.0056\end{array}$

$\begin{array}{rrr}\hfill 89& \hfill 313.6528& \hfill 310.4886\\ \hfill 90& \hfill 318.1526& \hfill 314.9829\\ \hfill 91& \hfill 322.6635& \hfill 319.4882\\ \hfill 92& \hfill 327.1853& \hfill 324.0045\end{array}$

$\begin{array}{rrr}\hfill 93& \hfill 331.7179& \hfill 328.5318\\ \hfill 94& \hfill 336.2612& \hfill 333.0697\\ \hfill 95& \hfill 340.8151& \hfill 337.6183\\ \hfill 96& \hfill 345.3794& \hfill 342.1774\end{array}$

$\begin{array}{rrr}\hfill 97& \hfill 349.9541& \hfill 346.747\\ \hfill 98& \hfill 354.5391& \hfill 351.3268\\ \hfill 99& \hfill 359.1342& \hfill 355.9169\\ \hfill 100& \hfill 363.7394& \hfill 360.517\end{array}$

For $N\ge 90$ Stirling’s approximation agrees with the true value to within $1\%$. These values are calculated using the percentage difference between the $\ln N!$ and $N\ln N-N$ for values $N=1$ to $100$.

Conclusion

For $N\ge 90$ Stirling’s approximation agrees with the true value to within $1\%$.