
Useful limits
Storyboard 
There are several approaches that occur when the number of cases / events is large.
ID:(1557, 0)

Sterling approximation
Equation 
James Stirling showed that the logarithm of the factorial function for large numbers can be approximated by
so you can approximate it by
\ln u!\sim\ln\sqrt{2\pi u} + u\ln u - u |
ID:(4737, 0)

Factorial according to Sterling's approximation
Equation 
Since the logarithm of the factorial according to Stirling can be approximated by
\ln u!\sim\ln\sqrt{2\pi u} + u\ln u - u |
the factorial itself can be estimated for large numbers by
u!\sim\sqrt{2\pi u}\left(\displaystyle\frac{u}{e}\right)^u |
ID:(8966, 0)

Taylor of \ln(1+u)
Equation 
If it is developed around
\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3) |
ID:(9000, 0)

Taylor series reformulation of \ln(1+u)
Equation 
With Taylor's development of
\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3) |
can be estimated
1+u\sim e^{u-\frac{1}{2}u^2} |
ID:(9001, 0)

Definition of the exponential function
Equation 
The exponential function is defined by the limit
so you can approximate
e^z\sim\left(1+\displaystyle\frac{z}{u}\right)^u |
ID:(8967, 0)

0
Video
Video: Useful limits