Twierdzenie Cauchy’ego-Hadamarda mówi o sposobie obliczania promienia zbieżności szeregu potęgowego .
Mamy szereg potęgowy
∑
n
=
1
∞
a
n
(
x
−
x
0
)
n
,
{\displaystyle \sum _{n=1}^{\infty }a_{n}(x-x_{0})^{n},}
(
x
0
−
R
,
x
0
+
R
)
.
{\displaystyle (x_{0}-R,x_{0}+R).}
R
{\displaystyle R}
R
=
{
∞
,
λ
=
0
0
,
λ
=
∞
1
λ
,
0
<
λ
<
∞
,
{\displaystyle R={\begin{cases}\infty ,&\lambda =0\\0,&\lambda =\infty \\{\frac {1}{\lambda }},&0<\lambda <\infty \end{cases}},}
gdzie
λ
=
lim sup
n
→
∞
|
a
n
|
n
.
{\displaystyle \lambda =\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}.}
Niech
x
∈
R
{\displaystyle x\in \mathbb {R} }
d
x
:=
lim sup
n
→
∞
|
a
n
(
x
−
x
0
)
n
|
n
=
λ
|
x
−
x
0
|
.
{\displaystyle d_{x}:=\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}(x-x_{0})^{n}|}}=\lambda |x-x_{0}|.}
kryterium Cauchy’ego mamy:
jeżeli
d
x
<
1
{\displaystyle d_{x}<1}
∑
a
n
(
x
−
x
0
)
n
{\displaystyle \sum a_{n}(x-x_{0})^{n}}
x
∈
P
,
{\displaystyle x\in P,}
jeżeli
d
x
>
1
{\displaystyle d_{x}>1}
∑
a
n
(
x
−
x
0
)
n
{\displaystyle \sum a_{n}(x-x_{0})^{n}}
x
∉
P
.
{\displaystyle x\notin P.}
Zauważamy, że
d
x
<
1
⇔
|
x
−
x
0
|
<
1
λ
{\displaystyle d_{x}<1\Leftrightarrow |x-x_{0}|<{\frac {1}{\lambda }}}
λ
{\displaystyle \lambda }
Jeżeli:
λ
=
0
⇒
∀
x
∈
R
{\displaystyle \lambda =0\Rightarrow \forall _{x\in \mathbb {R} }}
d
x
=
0
,
{\displaystyle d_{x}=0,}
d
x
<
1
,
x
∈
P
,
x
∈
R
,
{\displaystyle d_{x}<1,x\in P,x\in \mathbb {R} ,}
P
=
R
,
{\displaystyle P=\mathbb {R} ,}
λ
=
∞
⇒
{\displaystyle \lambda =\infty \Rightarrow }
x
≠
x
0
{\displaystyle x\neq x_{0}}
d
x
=
∞
,
{\displaystyle d_{x}=\infty ,}
d
x
>
1
,
{\displaystyle d_{x}>1,}
P
=
{
x
0
}
,
R
=
0
{\displaystyle P=\{x_{0}\},R=0}
0
<
λ
<
∞
{\displaystyle 0<\lambda <\infty }
d
x
<
1
⇔
|
x
−
x
0
|
<
1
λ
.
{\displaystyle d_{x}<1\Leftrightarrow |x-x_{0}|<{\frac {1}{\lambda }}.}
|
x
−
x
0
|
<
1
λ
⇒
x
∈
P
{\displaystyle |x-x_{0}|<{\frac {1}{\lambda }}\Rightarrow x\in P}
|
x
−
x
0
|
⩽
R
⇒
1
λ
⩽
R
.
{\displaystyle |x-x_{0}|\leqslant R\Rightarrow {\frac {1}{\lambda }}\leqslant R.}
1
λ
<
R
.
{\displaystyle {\frac {1}{\lambda }}<R.}
kresu górnego
∃
x
1
∈
P
|
x
1
−
x
0
|
>
1
λ
.
{\displaystyle \exists _{x_{1}\in P}|x_{1}-x_{0}|>{\frac {1}{\lambda }}.}
d
x
>
1
,
{\displaystyle d_{x}>1,}
∑
a
n
(
x
1
−
x
0
)
n
{\displaystyle \sum a_{n}(x_{1}-x_{0})^{n}}
x
1
∈
P
.
{\displaystyle x_{1}\in P.}
R
=
1
λ
.
{\displaystyle R={\frac {1}{\lambda }}.}
![{\displaystyle \lambda =\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf2f040bea24e44992a02aabf13e2b860e53fe6)
![{\displaystyle d_{x}:=\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}(x-x_{0})^{n}|}}=\lambda |x-x_{0}|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05fc64b52f589dd527177370bb9064d384b7bbd6)
