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\end_header

\begin_body

\begin_layout Title
List s formulami za 1.
 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

\begin_layout Plain Layout


\backslash
newcommand
\backslash
euler{e}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{multicols}{2}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\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
<lyxtabular version="3" rows="4" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
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</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sin$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\cos$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\tan$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $30^{\circ}=\frac{\pi}{6}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{\sqrt{3}}{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{\sqrt{3}}{3}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $45^{\circ}=\frac{\pi}{4}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{\sqrt{2}}{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{\sqrt{2}}{2}$
\end_inset


\end_layout

\end_inset
</cell>
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\begin_inset Text

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1
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</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $60^{\circ}=\frac{\pi}{3}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{\sqrt{3}}{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sqrt{3}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\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
Srečno!
\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
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