Answer
Five terms
Work Step by Step
We are given that $f(x)=\ln (1+x)$
and $f^a(x)=(-1)^{a+1}(a-1)!(1+x)^{-a}$
Check the Taylor inequality at $x=0.1$
Now, $|R_n(x)|\leq \dfrac{M}{(n+1)!}|x-a|^{n+1}$
This gives: $|R_5(x)|\leq \dfrac{n!}{(n+1)!}|0.4-0|^{n+1}\lt 0.001$
The above inequality can be rearranged as:
$|R_5(x)|\leq \dfrac{0.4^{n+1}}{(n+1)}\lt 0.001$
For the value of $n=5$:
$\dfrac{0.4^{5+1}}{(5+1)} \approx 0.00068\lt 0.001$
This shows that the inequality need five terms.
