Druyd:Knowledge

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 $1+u$

$n!$

Factorial $n!$

$n$

Number

$u$

Parameter $u$

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

$\ln u!\sim\ln\sqrt{2\pi u} + u\ln u - u$

\ln u!\sim\ln\sqrt{2\pi u} + u\ln u - u

$u!\sim\sqrt{2\pi u}\left(\displaystyle\frac{u}{e}\right)^u$

u!\sim\sqrt{2\pi u}\left(\displaystyle\frac{u}{e}\right)^u

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

e^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)

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

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

Examples

Sterling approximation

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

Factorial according to Sterling's approximation

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

Taylor of $\ln(1+u)$

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

equation

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

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

equation=9000

can be estimated

equation

Definition of the exponential function

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)