Mocninne rady
Power series
Mocninnou radu definujeme v oboru komplexnich cisel jako
kde z , s jsou komplexni cisla a ( ) je posloupnost komplexnich cisel. s se nazyva stred, jsou koeficienty mocninne rady. Bude nas zajimat, pro jaka z rada konverguje. Plati veta, ze pro kazdou mocninnou radu existuje jednoznacne urcene kladne realne cislo (nebo ) R takove, ze rada konverguje pro vsechna z pro nez a diverguje pro vsechna z pro nez > R. Toto cislo nazveme polomer konvergence .
Pokud existuje limita posloupnosti , je polomer konvergence . Je-li L=0, je R= , je-li L= , je R=0.
Power serie is defined in complex numbers as
where z , s are complex numbers and ( ) is complex serie. s is called center, coefficients of the power serie. The question is for what values of z does the serie converg. It can be proven that every power serie has associated unique positive real number (or ) R such that the serie convergs for every z such that and divergs for every z such that > R. We'll call this number the convergence radius .
Let's assume that the following limit exists: . Convergence radius can then be expressed as . If L=0 then R= , if L= then R=0.
Podivejme se, jak se chova mocinna rada :
Let's see how does this power serie behave: :
Dle teorie tedy tato rada konverguje pro z<1, coz odpovida i udajum na animovanem grafu.
According to the theory, this serie convergs for z<1, which is indeed what we can see at the plot.
Exponenciela:
The exponential function:
Nasledujici rada ma polomer konvergence 1 a obor konvergenze nelze rozsirit, rada osciluje ve vsech bodech jednotkove kruznice:
A tato rada ma polomer konvergence e:
Tato rada nikde nekonverguje: