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# Series 

sum mean

## Arithmetic Sum

$$\sum \limits_{k=1}^{n} k = \frac{n (n+1)}{2}$$

with $k,n \in \mathbb{N}$

## Geometric Sum

$$\sum \limits_{k=0}^{n} q^k = \frac{1 - q^{n+1}}{1-q}$$

with $k,n \in \mathbb{N}$ and $q \in \mathbb{Z}$

## Exponential row

$$\sum\limits_{n = 0}^{\infty} \frac{{\boldsymbol{z}}^n}{n!} = e^{{\boldsymbol{z}}}$$

with $n \in \mathbb{N}$ and ${\boldsymbol{z}} \in \mathbb{C}$

## Taylorpolynom

$T_{m,f,x_0}(x)$ is a row for $m \rightarrow \infty$.

$$_{m,f,x_0}(x)= \sum_{i=0}^{m} \underbrace{\frac{f^{(i)}(x_0)}{i!}}_{a_i} (x-x_0)^i }$$

with $x \in {\mathop{\mathbb R}}$, function $f$, and order $m \in {\mathop{\mathbb N}}$.

Konvergenzradius $R={\tiny\begin{matrix}\\ \normalsize \lim \\[1em] \scriptsize i\rightarrow \infty \end{matrix}} {\left\vert{\frac{a_i}{a_{i+1}}}\right\vert}$

## Taylor series

$$f({\boldsymbol{z}}) = \sum\limits_{k = 0}^{\infty} \frac{{\boldsymbol{f}}^{(k)}\left({\boldsymbol{z}}_0\right)}{k!} ({\boldsymbol{z}} - {\boldsymbol{z}}_0)^k$$

$f({\boldsymbol{z}})$ Taylor series Conditions
$e^{\boldsymbol{z}}$ $\sum_{n=0}^\infty \frac{{\boldsymbol{z}}^n}{n!}$ $\forall {\boldsymbol{z}} \in {\mathop{\mathbb C}}$
$\ln({\boldsymbol{z}})$ $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}({\boldsymbol{z}}-1)^n$ $0\lt{\boldsymbol{z}}\le2$
$\frac{1}{1-{\boldsymbol{z}}}$ $\sum^\infty_{n=0} {\boldsymbol{z}}^n$ ${\left\vert{{\boldsymbol{z}}}\right\vert} \lt 1$
$\sin {\boldsymbol{z}}$ $\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} {\boldsymbol{z}}^{2n+1}$ $\forall {\boldsymbol{z}} \in {\mathop{\mathbb C}}$
$\cos {\boldsymbol{z}}$ $\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} {\boldsymbol{z}}^{2n}$ $\forall {\boldsymbol{z}} \in {\mathop{\mathbb C}}$
$\sinh {\boldsymbol{z}}$ $\sum^{\infty}_{n=0} \frac{{\boldsymbol{z}}^{2n+1}}{(2n+1)!}$ $\forall {\boldsymbol{z}} \in {\mathop{\mathbb C}}$
$\cosh {\boldsymbol{z}}$ $\sum^{\infty}_{n=0} \frac{{\boldsymbol{z}}^{2n}}{(2n)!}$ $\forall {\boldsymbol{z}} \in {\mathop{\mathbb C}}$
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