Answer
Four Terms
Work Step by Step
Here, we have $f(x)=e^x \implies f^n(x)=e^x$
Now, we need to check the Taylor inequality at $x=0.1$
Now, $|R_n(x)|\leq \dfrac{M}{(n+1)!}|x-a|^{n+1}$
and $|R_2(x)|\leq \dfrac{e^{0.1}}{(2+1)!}|0.1-0|^{2+1}\approx 0.0002$
$|R_3(x)|\leq \dfrac{e^{0.1}}{(3+1)!}|0.1-0|^{3+1}\approx 0.000005$
We can see that the remainder is $|R_3(x)| \lt 0.00001$, thus we need four terms.