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