Answer
The fifth degree Taylor polynomial approximates $f(5)$ with error less than $0.0002$.
Work Step by Step
Here, we have $f^n(x)=\dfrac{(-1)^{n}n!}{3^n(n+1)}$ , $a=4$
Now, we need to check the fifth degree Taylor polynomial $T_5(x)$ when $n-5$
$f(5)=\dfrac{(-1)^{6}6!}{3^6(6+1)}=\dfrac{80}{567}$
For the next term, we have
$(\dfrac{80}{567}) \times \dfrac{(x-4)^{6}}{6!}=\dfrac{(x-4)^6}{5103}$
The absolute value will be
$f(5)=|\dfrac{(5-4)^{6}}{5103}|=\dfrac{1}{5103} \approx 0.000196\lt 0.0002$
Hence, it has been proved that the fifth degree Taylor polynomial approximates $f(5)$ with error less than $0.0002$.
