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
\(e^\cx{z}\)	\(\sum_{n=0}^\infty \frac{\cx{z}^n}{n!}\)	\(\forall \cx{z} \in \C\)
\(\ln(\cx{z})\)	\(\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(\cx{z}-1)^n\)	\(0<\cx{z}\le2\)
\(\frac{1}{1-\cx{z}}\)	\(\sum^\infty_{n=0} \cx{z}^n\)	\(\abs{\cx{z}} < 1\)
\(\sin \cx{z}\)	\(\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} \cx{z}^{2n+1}\)	\(\forall \cx{z} \in \C\)
\(\cos \cx{z}\)	\(\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} \cx{z}^{2n}\)	\(\forall \cx{z} \in \C\)
\(\sinh \cx{z}\)	\(\sum^{\infty}_{n=0} \frac{\cx{z}^{2n+1}}{(2n+1)!}\)	\(\forall \cx{z} \in \C\)
\(\cosh \cx{z}\)	\(\sum^{\infty}_{n=0} \frac{\cx{z}^{2n}}{(2n)!}\)	\(\forall \cx{z} \in \C\)