Exponential series/R/Estimate for remainder/Exercise

For ${}N\in \mathbb {N}$ and ${}x\in \mathbb {R}$ let

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

be the remainder of the exponential series. Prove that for

${}\vert {x}\vert \leq 1+{\frac {1}{2}}N\,$

the remainder term estimate

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

holds.

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