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sum mean

Arithmetic Sum

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

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

Geometric Sum

k=0nqk=1qn+11q\sum \limits_{k=0}^{n} q^k = \frac{1 - q^{n+1}}{1-q}

with k,nk,n \in \mathbb{N} and qq \in \mathbb{Z}

Exponential row

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

with nn \in \mathbb{N} and 𝐳\boldsymbol{z} \in \mathbb{C}

Taylorpolynom

Tm,f,x0(x)T_{m,f,x_0}(x) is a row for mm \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 ff, and order $m \in \N$.

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(𝐳)=k=0𝐟(k)(𝐳0)k!(𝐳𝐳0)kf(\boldsymbol{z}) = \sum\limits_{k = 0}^{\infty} \frac{\boldsymbol{f}^{(k)}\left(\boldsymbol{z}_0\right)}{k!} (\boldsymbol{z} - \boldsymbol{z}_0)^k

f(𝐳)f(\boldsymbol{z}) Taylor series Conditions
e𝐳e^\boldsymbol{z} n=0𝐳nn!\sum_{n=0}^\infty \frac{\boldsymbol{z}^n}{n!} $\forall \boldsymbol{z} \in \C$
ln(𝐳)\ln(\boldsymbol{z}) n=1(1)n+1n(𝐳1)n\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(\boldsymbol{z}-1)^n 0<𝐳20\lt\boldsymbol{z}\le2
11𝐳\frac{1}{1-\boldsymbol{z}} n=0𝐳n\sum^\infty_{n=0} \boldsymbol{z}^n |𝐳|<1\left\vert{\boldsymbol{z}}\right\vert \lt 1
sin𝐳\sin \boldsymbol{z} n=0(1)n(2n+1)!𝐳2n+1\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} \boldsymbol{z}^{2n+1} $\forall \boldsymbol{z} \in \C$
cos𝐳\cos \boldsymbol{z} n=0(1)n(2n)!𝐳2n\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} \boldsymbol{z}^{2n} $\forall \boldsymbol{z} \in \C$
sinh𝐳\sinh \boldsymbol{z} n=0𝐳2n+1(2n+1)!\sum^{\infty}_{n=0} \frac{\boldsymbol{z}^{2n+1}}{(2n+1)!} $\forall \boldsymbol{z} \in \C$
cosh𝐳\cosh \boldsymbol{z} n=0𝐳2n(2n)!\sum^{\infty}_{n=0} \frac{\boldsymbol{z}^{2n}}{(2n)!} $\forall \boldsymbol{z} \in \C$