Question

(a) Find the percent error in Stirling’s approximation for z = 10 ?

(b)What is the smallest integer z such that the error is less than 1%?

Short answer

(a) Error of Stirling's approximation for z =10 is 13.75%.

(b) Error of Stirling's approximation is less than 1% for z = 90.

Step-by-step solution

Define the Stirling’s approximation

• Stirling's approximation (or Stirling's formula) is a factorial approximation in mathematics.
• It is a good approximation, yielding accurate results even for low displaystyle nn values. It takes its name from James Stirling.

Calculating the percent error

(a)

For z = 10 we have

$$\begin{gathered} \mathrm{In}(10!)=15.1044 \\ 10\mathrm{In}\left(10\right)-10=13.0259 \\ \%\mathrm{Error}=\frac{15.1044-13.0259}{15.1044}13.76\% \end{gathered}$$

Therefore the percent error in Stirling’s approximation is 13.76% .

Calculating the smallest integer

(b)

% Error is calculated by:

$$\begin{gathered} \frac{In(z!)zIn\left(z\right)+z}{In(2!)}.100 \\ \begin{array}{cc}z& \%Error\\ 21& 5.67\\ 100& 0.89\\ 90& 0.996\\ 85& 1.06\\ 89& 1.009\end{array} \end{gathered}$$

Therefore we see that for z = 90 and more error is less than 1%.