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theorems-ams
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\begin_body
\begin_layout Title
List s formulami za 2.
kolokvij Analize 1
\end_layout
\begin_layout Author
\noun on
Anton Luka Šijanec
\end_layout
\begin_layout Date
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
today
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
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\backslash
newcommand
\backslash
euler{e}
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status open
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\backslash
setlength{
\backslash
columnseprule}{0.2pt}
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begin{multicols}{2}
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\end_inset
\begin_inset Formula $\log_{a}1=0$
\end_inset
,
\begin_inset Formula $\log_{a}a=1$
\end_inset
,
\begin_inset Formula $\log_{a}a^{x}=x$
\end_inset
,
\begin_inset Formula $a^{\log_{a}x}=x$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\log_{a}x^{n}=n\log_{a}x$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $D=b^{2}-4ac$
\end_inset
,
\begin_inset Formula $x_{1,2}=\frac{-b\pm\sqrt{D}}{2a}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $zw=\left(ac-bd\right)+\left(ad+bc\right)i$
\end_inset
,
\begin_inset Formula $\vert zw\vert=\vert z\vert\vert w\vert$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\arg\left(zw\right)=\arg z+\arg w$
\end_inset
(kot)
\end_layout
\begin_layout Standard
\begin_inset Formula $z\overline{z}=a^{2}-\left(bi\right)^{2}=a^{2}+b^{2}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(\cos\phi+i\sin\phi\right)$
\end_inset
\begin_inset Formula $\left(\cos\psi+i\sin\psi\right)=\cos\left(\phi+\psi\right)+i\sin\left(\phi+\psi\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $z^{2}=a^{2}+2abi-b^{2}$
\end_inset
,
\begin_inset Formula $z^{3}=a^{3}-3ab^{2}+\left(3a^{2}b-b^{3}\right)i$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $(a+b)^{n}=\sum_{k=0}^{n}{n \choose k}ab^{n-k}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $z^{n}=r^{3}\left(\cos\left(3\phi\right)+i\sin\left(3\phi\right)\right)$
\end_inset
,
\begin_inset Formula $\phi=\arctan\frac{\Im z}{\Re z}$
\end_inset
\end_layout
\begin_layout Standard
Odprta množica ne vsebuje robnih točk.
Zaprta vsebuje vse.
\end_layout
\begin_layout Standard
\begin_inset Formula $\sin\left(x\pm y\right)=\sin x\cdot\cos y\pm\sin y\cdot\cos x$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\cos\left(x\pm y\right)=\cos x\cdot\cos y\mp\sin y\cdot\sin x$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\tan\left(x\pm y\right)=\frac{\tan x\pm\tan y}{1\text{\ensuremath{\mp\tan}x\ensuremath{\cdot\tan y}}}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $a_{n}$
\end_inset
je konv.
\begin_inset Formula $\Longleftrightarrow$
\end_inset
\begin_inset Formula $\forall\varepsilon>0:\exists n_{0}\ni:\forall n,m:n_{0}<n<m\wedge\vert a_{n}-a_{m}\vert<\varepsilon$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\euler^{1/k}\coloneqq\lim_{n\to\infty}\left(1+\frac{1}{nk}\right)^{n}$
\end_inset
\end_layout
\begin_layout Standard
Vrsta je konv., če je konv.
njeno zap.
delnih vsot.
\end_layout
\begin_layout Standard
\begin_inset Formula $s_{n}=\begin{cases}
\frac{1-q^{n+1}}{1-q}; & q\not=1\\
n+1; & q=1
\end{cases}$
\end_inset
.
Geom.
vrsta konv.
\begin_inset Formula $\Longleftrightarrow q\in\left(-1,1\right)$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Primerjalni krit.
\series default
:
\begin_inset Formula $\sum_{1}^{\infty}a_{k}$
\end_inset
konv.
\begin_inset Formula $\wedge$
\end_inset
\begin_inset Formula $b_{k}\leq a_{k}$
\end_inset
za
\begin_inset Formula $k>n_{0}$
\end_inset
\begin_inset Formula $\wedge$
\end_inset
vrsti sta navzdol omejeni
\begin_inset Formula $\Longrightarrow$
\end_inset
\begin_inset Formula $\sum_{1}^{\infty}b_{k}$
\end_inset
konv.
\begin_inset Formula $\sum_{1}^{\infty}a_{k}$
\end_inset
rečemo
\shape italic
majoranta
\shape default
.
\end_layout
\begin_layout Standard
\series bold
Kvocientni
\series default
:
\begin_inset Formula $a_{k}>0$
\end_inset
,
\begin_inset Formula $D_{n}\coloneqq\frac{a_{n}+1}{a_{n}}$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall n<n_{0}:D_{n}\in\left(0,1\right)\Longrightarrow\sum_{1}^{\infty}a_{k}<\infty$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall n<n_{0}:D_{n}\geq1\Longrightarrow\sum_{1}^{\infty}a_{k}=\infty$
\end_inset
\end_layout
\begin_layout Itemize
Če
\begin_inset Formula $\exists D\coloneqq\lim_{n\to\infty}D_{n}$
\end_inset
:
\begin_inset Formula $\vert D\vert<1\Longrightarrow$
\end_inset
konv.,
\begin_inset Formula $\vert D\vert>1\Longrightarrow div.$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Korenski
\series default
: Kot Kvocientni, le da
\begin_inset Formula $D_{n}\coloneqq\sqrt[n]{a_{n}}$
\end_inset
.
\end_layout
\begin_layout Standard
\series bold
Leibnizov
\series default
:
\begin_inset Formula $a_{n}\to0\Longrightarrow\sum_{1}^{\infty}\left(\left(-1\right)^{k}a_{k}\right)<\infty$
\end_inset
\end_layout
\begin_layout Standard
Absolutna konvergenca
\begin_inset Formula $\left(\sum_{1}^{\infty}\vert a_{n}\vert<\infty\right)$
\end_inset
\begin_inset Formula $\Longrightarrow$
\end_inset
konvergenca
\end_layout
\begin_layout Standard
Pri konv.
po točkah je
\begin_inset Formula $n_{0}$
\end_inset
odvisen od
\begin_inset Formula $x$
\end_inset
, pri enakomerni ni.
\end_layout
\begin_layout Standard
Potenčna vrsta:
\begin_inset Formula $\sum_{j=1}^{\infty}b_{j}x^{j}$
\end_inset
.
\begin_inset Formula $R^{-1}=\limsup_{k\to\infty}\sqrt[k]{\vert b_{k}\vert}$
\end_inset
.
\begin_inset Formula $\vert x\vert<R\Longrightarrow$
\end_inset
abs.
konv.,
\begin_inset Formula $\vert x\vert>R\Longrightarrow$
\end_inset
divergira
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{x\to a}\left(\alpha f\left(x\right)\right)=\alpha\lim_{x\to a}f\left(x\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Tabular
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\begin_layout Plain Layout
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\begin_inset Text
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1
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\begin_layout Standard
Krožnica:
\begin_inset Formula $\left(x-p\right)^{2}+\left(y-q\right)^{2}=r^{2}$
\end_inset
\end_layout
\begin_layout Standard
Elipsa:
\begin_inset Formula $\frac{\left(x-p\right)^{2}}{a^{2}}+\frac{\left(y-q\right)^{2}}{b^{2}}=1$
\end_inset
\end_layout
\begin_layout Standard
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,
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\begin_inset Text
\begin_layout Plain Layout
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\end_inset
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\begin_inset Text
\begin_layout Plain Layout
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\end_inset
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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\end_inset
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\begin_layout Plain Layout
\begin_inset Formula $\text{arccot\,}x$
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<row>
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konveksna na
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konkavna na
\begin_inset Formula $I$
\end_inset
\begin_inset Formula
\[
ab>0\wedge a<b\Leftrightarrow a^{-1}>b^{-1},\quad ab<0\wedge a<b\Leftrightarrow a^{-1}<b^{-1}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\lim_{x\to0}\frac{\sin x}{x}=1\quad\quad\tan\phi=\left|\frac{k_{1}-k_{2}}{1+k_{1}k_{2}}\right|
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\lim_{x\to0}x\ln x=0
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
f\text{ zv.+odv.@ }\left[a,b\right]\Rightarrow\exists\xi\in\left[a,b\right]\ni:f\left(b\right)-f\left(a\right)=f'\left(\xi\right)\left(b-a\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
T_{f,a,n}\left(x\right)=\sum_{k=0}^{n}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^{k}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f\text{\ensuremath{\in C^{n+1}}}$
\end_inset
na odprtem
\begin_inset Formula $I\subset\mathbb{R}\Rightarrow\forall a,x\in I\exists c\in\left(\min\left\{ a,x\right\} ,\max\left\{ a,x\right\} \right)\ni:f\left(x\right)-T_{f,a,n}\left(x\right)=R_{f,a,n}\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}$
\end_inset
\begin_inset Formula $\left(x-a\right)^{n+1}.\text{ Posledično velja tudi takale ocena:}$
\end_inset
\begin_inset Formula
\[
\exists M>0\forall x\in I:\left|f^{\left(n+1\right)}\right|\leq M\Rightarrow R_{f,a,n}\left(x\right)=\frac{M}{\left(n+1\right)!}\left|x-a\right|^{n+1}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
R=\lim_{n\to\infty}\left|\frac{c_{n}}{c_{n+1}}\right|,\quad R=\lim_{n\to\infty}\frac{1}{\sqrt[n]{\left|c_{n}\right|}}
\]
\end_inset
\end_layout
\begin_layout Standard
Zvezna
\begin_inset Formula $\text{f}$
\end_inset
na zaprtem intervalu
\begin_inset Formula $\left[a,b\right]$
\end_inset
doseže
\begin_inset Formula $\inf$
\end_inset
in
\begin_inset Formula $\sup$
\end_inset
, je omejena in doseže vse funkcijske vrednosti na
\begin_inset Formula $\left[f\left(a\right),f\left(b\right)\right]$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Formula $f:I\to\mathbb{R}$
\end_inset
je enakomerno zvezna na
\begin_inset Formula $I$
\end_inset
, če
\begin_inset Formula $\forall\varepsilon>0\exists\delta_{\left(\varepsilon\right)}>0\ni:\forall x,y\in I:\left|x-y\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(y\right)\right|<\varepsilon$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $f:I\to\mathbb{R}$
\end_inset
je zvezna na
\begin_inset Formula $I$
\end_inset
, če
\begin_inset Formula $\forall\varepsilon>0\forall x\in I\exists\delta_{\left(x,\varepsilon\right)}>0\ni:\forall x,y\in I:\left|x-y\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(y\right)\right|<\varepsilon$
\end_inset
\end_layout
\begin_layout Standard
Zvezna
\begin_inset Formula $f$
\end_inset
na kompaktni množici je enakomerno zvezna.
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
f'\left(x\right)=\lim_{x\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sinh x=\frac{e^{x}-e^{-x}}{2},\quad\cosh x=\frac{e^{x}+e^{-x}}{2}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{multicols}
\end_layout
\end_inset
\end_layout
\end_body
\end_document
