Answer
The fifth degree Taylor polynomial approximates $f(5)$ with error less than $0.0002$.
Work Step by Step
We are given that $f^n(x)=\dfrac{(-1)^{n}n!}{3^n(n+1)}$ , $a=4$
The fifth degree Taylor polynomial $T_5(x)$ when $n-5$ can be defined as:
$f(5)=\dfrac{(-1)^{6}6!}{3^6 \times (6+1)}=\dfrac{80}{567}$
and $f(6)=\dfrac{80}{567} \times [\dfrac{(x-4)^{6}}{6!}]=\dfrac{(x-4)^6}{5103}$
Therefore, the absolute value can be found as:
$f(5)=|\dfrac{(5-4)^{6}}{5103}|=\dfrac{1}{5103} \approx 0.000196\lt 0.0002$
Hence, this is proved that the fifth degree Taylor polynomial approximates $f(5)$ with error less than $0.0002$.
