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

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ID:(1557, 0)


Useful limits
Description

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

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$1+u$
1+u
Desarrollo $1+u$
-
$n!$
n!
Factorial $n!$
-
$n$
n
Number
-
$u$
u
Parameter $u$
-

Calculations

First, select the equation:   to ,  then, select the variable:   to 
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

Examples

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)

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)

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)

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)

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)

ID:(1557, 0)