# Exponentialreihe/C/Abschätzung für Restglied/Aufgabe/en

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For ${\displaystyle {}N\in \mathbb {N} }$ and ${\displaystyle {}z\in \mathbb {C} }$ let

${\displaystyle R_{N+1}(z)=\exp z-\sum _{n=0}^{N}{\frac {z^{n}}{n!}}=\sum _{n=N+1}^{\infty }{\frac {z^{n}}{n!}}}$

be the remainder of the exponential series. Prove that for ${\displaystyle {}\vert {z}\vert \leq 1+{\frac {1}{2}}N}$ the remainder term estimate

${\displaystyle \vert {R_{N+1}(z)}\vert \leq {\frac {2}{(N+1)!}}\vert {z}\vert ^{N+1}}$

holds.