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:
:
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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:
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Nasledujici rada ma polomer konvergence 1 a obor konvergenze nelze rozsirit, rada osciluje ve vsech bodech jednotkove kruznice:
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A tato rada ma polomer konvergence e:
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Tato rada nikde nekonverguje:
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