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Useful limits
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There are several approaches that occur when the number of cases / events is large.

>Model

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Sterling approximation
Equation

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James Stirling showed that the logarithm of the factorial function for large numbers can be approximated by

\ln u!=\ln\sqrt{2\pi u} + u\ln u - u+O(\ln u)

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

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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

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If it is developed around u=0 the logarithm of 1+u is obtained

$\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3)$

ID:(9000, 0)


Taylor series reformulation of $\ln(1+u)$
Equation

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With Taylor's development of \ln(1+u)

$\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3)$



can be estimated

$1+u\sim e^{u-\frac{1}{2}u^2}$

ID:(9001, 0)


Definition of the exponential function
Equation

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The exponential function is defined by the limit

e^z=\lim_{u\rightarrow\infty}\left(1+\displaystyle\frac{z}{u}\right)^u

so you can approximate

$e^z\sim\left(1+\displaystyle\frac{z}{u}\right)^u$

ID:(8967, 0)


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