mocninne_rady.html

Mocninne rady

Power series

Mocninnou radu definujeme v oboru komplexnich cisel jako

sum(a[n]*(z-s)^n,n = 0 .. infinity)

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 L = Limit(abs(a[n])^(1/n),n = infinity) , je polomer konvergence R = 1/L . Je-li L=0, je R= , je-li L= , je R=0.

Power serie is defined in complex numbers as

sum(a[n]*(z-s)^n,n = 0 .. infinity)

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: L = Limit(abs(a[n])^(1/n),n = infinity) . Convergence radius can then be expressed as R = 1/L . 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: :

Sum(n*z^n,n = 1 .. N) [Maple Plot]

Limit(abs(n)^(1/n),n = infinity) = 1

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:

[Maple Plot]

Tato rada nikde nekonverguje: