Answer
The fifth degree Taylor polynomial approximates $f(5)$ with error less than $0.0002$.
Work Step by Step
Given: $f^n(x)=\dfrac{(-1)^{n}n!}{3^n(n+1)}$ and $a=4$
This gives:
$f(5)=\dfrac{(-1)^{6}6!}{3^6(7)}=\dfrac{80}{567}$
Out next term would be:
$(\dfrac{80}{567}) \times \dfrac{(x-4)^{6}}{6!}=\dfrac{(x-4)^6}{5103}$
Now, the absolute value would become:
$f(5)=|\dfrac{(5-4)^{6}}{5103}|=\dfrac{1}{5103}$
or, $f(5) \approx 0.000196\lt 0.0002$
we can see that the fifth degree Taylor polynomial approximates $f(5)$ with error less than $0.0002$.
