# Series#

## Arithmetic Sum#

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

with $$k,n ∈ ℕ$$

## Geometric Sum#

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

with $$k,n ∈ ℕ$$ and $$q ∈ ℤ$$

## Exponential row#

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

with $$n ∈ ℕ$$ and $$\cx z ∈ ℂ$$

## 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 $$x \in \R$$, function $$f$$, and order $$m \in \N$$.

Konvergenzradius $$R=\underset{i\rightarrow \infty}{\lim} \abs{\frac{a_i}{a_{i+1}}}$$

## Taylor series#

$f(\cx z) = \sum\limits_{k = 0}^{\infty} \frac{\cx f^{(k)}\left(\cx z_0\right)}{k!} (\cx z - \cx z_0)^k$
$$f(\cx z)$$ Taylor series Conditions
$$e^\cx{z}$$ $$\sum_{n=0}^\infty \frac{\cx{z}^n}{n!}$$ $$\forall \cx{z} \in \C$$
$$\ln(\cx{z})$$ $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(\cx{z}-1)^n$$ $$0<\cx{z}\le2$$
$$\frac{1}{1-\cx{z}}$$ $$\sum^\infty_{n=0} \cx{z}^n$$ $$\abs{\cx{z}} < 1$$
$$\sin \cx{z}$$ $$\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} \cx{z}^{2n+1}$$ $$\forall \cx{z} \in \C$$
$$\cos \cx{z}$$ $$\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} \cx{z}^{2n}$$ $$\forall \cx{z} \in \C$$
$$\sinh \cx{z}$$ $$\sum^{\infty}_{n=0} \frac{\cx{z}^{2n+1}}{(2n+1)!}$$ $$\forall \cx{z} \in \C$$
$$\cosh \cx{z}$$ $$\sum^{\infty}_{n=0} \frac{\cx{z}^{2n}}{(2n)!}$$ $$\forall \cx{z} \in \C$$