Answer
Five Terms
Work Step by Step
Check the Taylor inequality at $x=0.1$ when $f(x)=\ln (1+x)$
$|R_5(x)|\leq \dfrac{n!}{(n+1)!}|0.4-0|^{n+1}\lt 0.001$
This gives: $|R_5(x)|\leq \dfrac{n!(0.4)^{n+1}}{(n+1)n!}\lt 0.001$
and $|R_5(x)|\leq \dfrac{(0.4)^{n+1}}{(n+1)}\lt 0.001$
When $n=5$, then we have
$\dfrac{0.4^{5+1}}{(5+1)} \lt 0.001$
or, $\dfrac{0.4^{6}}{(6)} \approx 0.00068\lt 0.001$
It has been shown that we need Five terms.