Exponentialreihe/Konvergiert nicht gleichmäßig/Aufgabe/2/Lösung

Wähle ${\displaystyle \epsilon =1}$, ${\displaystyle N\in \mathbb {N} }$ beliebig, und ${\displaystyle n\in \mathbb {N} \land n\geq N}$ mit ${\displaystyle \mathbb {R} \ni x={\sqrt[{n+1}]{2(n+1)!}}}$ .

${\displaystyle \Rightarrow x>0\Rightarrow \sum _{\text{k=0}}^{n}{\frac {x^{k}}{k!}}>0}$ und streng monoton wachsend.

${\displaystyle \Rightarrow |}$exp${\displaystyle (x)-\sum _{\text{k=0}}^{n}{\frac {x^{k}}{k!}}|=\sum _{\text{k=0}}^{\infty }{\frac {x^{k}}{k!}}-\sum _{\text{k=0}}^{n}{\frac {x^{k}}{k!}}=\sum _{\text{k=n+1}}^{\infty }{\frac {x^{k}}{k!}}>\sum _{\text{k=n+1}}^{n+1}{\frac {x^{k}}{k!}}}$ =

${\displaystyle ={\frac {x^{n+1}}{(n+1)!}}={\frac {{\sqrt[{n+1}]{(2(n+1)!))}}^{n+1}}{(n+1)!}}={\frac {2(n+1)!}{(n+1)!}}=2>\epsilon }$

${\displaystyle \Rightarrow }$

${\displaystyle \mathbb {R} }$

${\displaystyle \Rightarrow }$

${\displaystyle \mathbb {C} }$