Answer
Four terms
Work Step by Step
We are given that $f(x)=e^x$ and $f^n(x)=e^x$
Check the Taylor inequality at $x=0.1$
Since, we have $|R_n(x)|\leq \dfrac{M}{(n+1)!}|x-a|^{n+1}$
This gives: $|R_2(x)|\leq \dfrac{e^{(0.1)}}{(2+1)!}|0.1-0|^{(2+1)}\approx 0.0002$
and $|R_3(x)|\leq \dfrac{e^{(0.1)}}{(3+1)!}|0.1-0|^{(3+1)}\approx 0.000005$
It has been seen that the remainder $|R_3(x)| \lt 0.00001$ require four terms.