Dgdrxrt

I'm really struggling to prove the following claim and I was wondering whether anyone could help me.

Claim:$$ 0<|x| leq 10^-4 implies bigg fracxe^x-1 - frac1sum_k=1^3 fracx^k-1k! bigg leq 10^-12$$

My attempt at a proof: I have already proven the following inequality,
$$bigg frace^x-1x - sum_k=1^n fracx^k-1k! bigg leq frac(n+1)!$$
So let $n=3$, then
beginalign*
bigg fracxe^x-1 - frac1sum_k=1^3 fracx^k-1k! bigg &= bigg fracx(e^x-1)(sum_k=1^3 fracx^k-1k!) cdot left( frace^x-1x - sum_k=1^3 fracx^k-1k! right) bigg \
&= bigg fracx(e^x-1)(sum_k=1^3 fracx^k-1k!) bigg cdot bigg frace^x-1x - sum_k=1^3 fracx^k-1k! bigg \
&leq bigg fracx(e^x-1)(sum_k=1^3 fracx^k-1k!) bigg cdot fracx(3+1)! \
&leq bigg fracx(e^x-1)(sum_k=1^3 fracx^k-1k!) bigg cdot frac(10^-4)^3 e^10^-44! \
&= 10^-12 cdot bigg fracx(e^x-1)(sum_k=1^3 fracx^k-1k!) bigg cdot frac e^10^-44!
endalign* Hence, it suffices to show that for $0 < |x| leq 10^-4$,
beginalign*
f(x) := bigg fracx(e^x-1)(sum_k=1^3 fracx^k-1k!) bigg < frac4!e^10^-4
endalign*
Since $f(x)$ is monotonically decreasing on the intervals $[-10^-4, 0)$ and $(0, 10^-4]$, we know that for $0 <|x| leq 10^-4$,
beginalign*
f(x) leq f left( -10^-4 right) approx 1.0001 < frac4!e^10^-4
endalign* Therefore, for $0 < |x| leq 10^-4$,
beginalign*
bigg fracxe^x-1 - frac1sum_k=1^3 fracx^k-1k! bigg leq 10^-12
endalign* $square$

My big problem here is that I don't actually know how to prove my claim that $f(x)$ is monotonically decreasing. Does anyone know how to prove this?

Or even better, does anyone know of a more direct proof that avoids this problem altogether?

Thanks!

Write $|fracxe^x-1 - fracxsum_k=1^3 fracx^kk!|
= |x||fracsum_k=1^3 fracx^kk! - (e^x-1)(e^x-1)sum_k=1^3 fracx^kk!| $

You can use the $|x|<10^-4$ bound given, and also instead an easier inequality for the numerator using the series sum for $e^x$. Then it is clear that the denominator is obviously increasing in $x$.

So I guess this comes under "direct proof". I think your function $f(x)$ is probably decreasing though.

For $|x|<epsilon=10^-4$, by the mean value theorem $$left|frace^x-1xright|=e^yge e^-epsilonge1-epsilon,$$ and $$1+fracx2+fracx^26ge1-fracepsilon2,$$ so it follows that $$left|fracxe^x-1frac11+fracx2+fracx^26right|lefrac1(1-epsilon)^2le2lefrac4!e^epsilon$$