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Useful limits
Storyboard

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

>Model

ID:(1557, 0)


Useful limits
Storyboard

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 
\ln u!\sim\ln\sqrt{2\pi u} + u\ln u - uu!\sim\sqrt{2\pi u}\left(\displaystyle\frac{u}{e}\right)^ue^z\sim\left(1+\displaystyle\frac{z}{u}\right)^u\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3)\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3)1+un!nu
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used
\ln u!\sim\ln\sqrt{2\pi u} + u\ln u - uu!\sim\sqrt{2\pi u}\left(\displaystyle\frac{u}{e}\right)^ue^z\sim\left(1+\displaystyle\frac{z}{u}\right)^u\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3)\ln(1+u)= u-\displaystyle\frac{1}{2}u^2+O(u^3)1+un!nu


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

equation

Since the logarithm of the factorial according to Stirling can be approximated by

equation=4737

the factorial itself can be estimated for large numbers by

equation

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

equation

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

equation=9000

can be estimated

equation

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

equation

>Model

ID:(1557, 0)